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Adequation

Adequation Ad`e*qua"tion, n. [L. adaequatio.] The act of equalizing; act or result of making adequate; an equivalent. [Obs.] --Bp. Barlow.

Adequation Ad`e*qua"tion, n. [L. adaequatio.] The act of equalizing; act or result of making adequate; an equivalent. [Obs.] --Bp. Barlow.

an equation

Discussion Dis*cus"sion, n. [L. discussio a shaking, examination, discussion: cf. F. discussion.] 1. The act or process of discussing by breaking up, or dispersing, as a tumor, or the like. 2. The act of discussing or exchanging reasons; examination by argument; debate; disputation; agitation. The liberty of discussion is the great safeguard of all other liberties. --Macaulay. Discussion of a problem or an equation (Math.), the operation of assigning different reasonable values to the arbitrary quantities and interpreting the result. --Math. Dict.

Discussion Dis*cus"sion, n. [L. discussio a shaking, examination, discussion: cf. F. discussion.] 1. The act or process of discussing by breaking up, or dispersing, as a tumor, or the like. 2. The act of discussing or exchanging reasons; examination by argument; debate; disputation; agitation. The liberty of discussion is the great safeguard of all other liberties. --Macaulay. Discussion of a problem or an equation (Math.), the operation of assigning different reasonable values to the arbitrary quantities and interpreting the result. --Math. Dict.

Biquadratic equation

Biquadratic Bi`quad*rat"ic, a. [Pref. bi- + quadratic: cf. F. biquadratique.] (Math.) Of or pertaining to the biquadrate, or fourth power. Biquadratic equation (Alg.), an equation of the fourth degree, or an equation in some term of which the unknown quantity is raised to the fourth power. Biquadratic root of a number, the square root of the square root of that number. Thus the square root of 81 is 9, and the square root of 9 is 3, which is the biquadratic root of 81. Hutton.

Biquadratic Bi`quad*rat"ic, a. [Pref. bi- + quadratic: cf. F. biquadratique.] (Math.) Of or pertaining to the biquadrate, or fourth power. Biquadratic equation (Alg.), an equation of the fourth degree, or an equation in some term of which the unknown quantity is raised to the fourth power. Biquadratic root of a number, the square root of the square root of that number. Thus the square root of 81 is 9, and the square root of 9 is 3, which is the biquadratic root of 81. Hutton.

Construction of an equation

Construction Con*struc"tion, n. [L. constructio: cf. F. construction.] 1. The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition. 2. The form or manner of building or putting together the parts of anything; structure; arrangement. An astrolabe of peculiar construction. --Whewell. 3. (Gram.) The arrangement and connection of words in a sentence; syntactical arrangement. Some particles . . . in certain constructions have the sense of a whole sentence contained in them. --Locke. 4. The method of construing, interpreting, or explaining a declaration or fact; an attributed sense or meaning; understanding; explanation; interpretation; sense. Any person . . . might, by the sort of construction that would be put on this act, become liable to the penalties of treason. --Hallam. Strictly, the term [construction] signifies determining the meaning and proper effect of language by a consideration of the subject matter and attendant circumstances in connection with the words employed. --Abbott. Interpretation properly precedes construction, but it does not go beyond the written text. --Parsons. Construction of an equation (Math.), the drawing of such lines and figures as will represent geometrically the quantities in the equation, and their relations to each other. Construction train (Railroad), a train for transporting men and materials for construction or repairs.

Construction Con*struc"tion, n. [L. constructio: cf. F. construction.] 1. The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition. 2. The form or manner of building or putting together the parts of anything; structure; arrangement. An astrolabe of peculiar construction. --Whewell. 3. (Gram.) The arrangement and connection of words in a sentence; syntactical arrangement. Some particles . . . in certain constructions have the sense of a whole sentence contained in them. --Locke. 4. The method of construing, interpreting, or explaining a declaration or fact; an attributed sense or meaning; understanding; explanation; interpretation; sense. Any person . . . might, by the sort of construction that would be put on this act, become liable to the penalties of treason. --Hallam. Strictly, the term [construction] signifies determining the meaning and proper effect of language by a consideration of the subject matter and attendant circumstances in connection with the words employed. --Abbott. Interpretation properly precedes construction, but it does not go beyond the written text. --Parsons. Construction of an equation (Math.), the drawing of such lines and figures as will represent geometrically the quantities in the equation, and their relations to each other. Construction train (Railroad), a train for transporting men and materials for construction or repairs.

Cubic equation

Cubic Cu"bic (k?"b?k), Cubical Cu"bic*al (-b?-kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See Cube.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. 2. (Crystallog.) Isometric or monometric; as, cubic cleavage. See Crystallization. Cubic equation, an equation in which the highest power of the unknown quantity is a cube. Cubic foot, a volume equivalent to a cubical solid which measures a foot in each of its dimensions. Cubic number, a number produced by multiplying a number into itself, and that product again by the same number. See Cube. Cubical parabola (Geom.), two curves of the third degree, one plane, and one on space of three dimensions.

Cubic Cu"bic (k?"b?k), Cubical Cu"bic*al (-b?-kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See Cube.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. 2. (Crystallog.) Isometric or monometric; as, cubic cleavage. See Crystallization. Cubic equation, an equation in which the highest power of the unknown quantity is a cube. Cubic foot, a volume equivalent to a cubical solid which measures a foot in each of its dimensions. Cubic number, a number produced by multiplying a number into itself, and that product again by the same number. See Cube. Cubical parabola (Geom.), two curves of the third degree, one plane, and one on space of three dimensions.

Determinate equations

Determinate De*ter"mi*nate, a. [L. determinatus, p. p. of determinare. See Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; established; definite. Quantity of words and a determinate number of feet. --Dryden. 2. Conclusive; decisive; positive. The determinate counsel and foreknowledge of God. --Acts ii. 23. 3. Determined or resolved upon. [Obs.] My determinate voyage. --Shak. 4. Of determined purpose; resolute. [Obs.] More determinate to do than skillful how to do. --Sir P. Sidney. Determinate inflorescence (Bot.), that in which the flowering commences with the terminal bud of a stem, which puts a limit to its growth; -- also called centrifugal inflorescence. Determinate problem (Math.), a problem which admits of a limited number of solutions. Determinate quantities, Determinate equations (Math.), those that are finite in the number of values or solutions, that is, in which the conditions of the problem or equation determine the number.

Determinate De*ter"mi*nate, a. [L. determinatus, p. p. of determinare. See Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; established; definite. Quantity of words and a determinate number of feet. --Dryden. 2. Conclusive; decisive; positive. The determinate counsel and foreknowledge of God. --Acts ii. 23. 3. Determined or resolved upon. [Obs.] My determinate voyage. --Shak. 4. Of determined purpose; resolute. [Obs.] More determinate to do than skillful how to do. --Sir P. Sidney. Determinate inflorescence (Bot.), that in which the flowering commences with the terminal bud of a stem, which puts a limit to its growth; -- also called centrifugal inflorescence. Determinate problem (Math.), a problem which admits of a limited number of solutions. Determinate quantities, Determinate equations (Math.), those that are finite in the number of values or solutions, that is, in which the conditions of the problem or equation determine the number.

Exponential equation

Exponential Ex`po*nen"tial, a. [Cf. F. exponentiel.] Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function. Exponential curve, a curve whose nature is defined by means of an exponential equation. Exponential equation, an equation which contains an exponential quantity, or in which the unknown quantity enters as an exponent. Exponential quantity (Math.), a quantity whose exponent is unknown or variable, as a^x. Exponential series, a series derived from the development of exponential equations or quantities.

Exponential Ex`po*nen"tial, a. [Cf. F. exponentiel.] Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function. Exponential curve, a curve whose nature is defined by means of an exponential equation. Exponential equation, an equation which contains an exponential quantity, or in which the unknown quantity enters as an exponent. Exponential quantity (Math.), a quantity whose exponent is unknown or variable, as a^x. Exponential series, a series derived from the development of exponential equations or quantities.

Identical equation

Identical I*den"tic*al, a. [Cf. F. identique. See Identity.] 1. The same; the selfsame; the very same; not different; as, the identical person or thing. I can not remember a thing that happened a year ago, without a conviction . . . that I, the same identical person who now remember that event, did then exist. --Reid. 2. Uttering sameness or the same truth; expressing in the predicate what is given, or obviously implied, in the subject; tautological. When you say body is solid, I say that you make an identical proposition, because it is impossible to have the idea of body without that of solidity. --Fleming. Identical equation (Alg.), an equation which is true for all values of the algebraic symbols which enter into it.

Identical I*den"tic*al, a. [Cf. F. identique. See Identity.] 1. The same; the selfsame; the very same; not different; as, the identical person or thing. I can not remember a thing that happened a year ago, without a conviction . . . that I, the same identical person who now remember that event, did then exist. --Reid. 2. Uttering sameness or the same truth; expressing in the predicate what is given, or obviously implied, in the subject; tautological. When you say body is solid, I say that you make an identical proposition, because it is impossible to have the idea of body without that of solidity. --Fleming. Identical equation (Alg.), an equation which is true for all values of the algebraic symbols which enter into it.

Inadequation

Inadequation In*ad`e*qua"tion, n. Want of exact correspondence. [Obs.] --Puller.

Inadequation In*ad`e*qua"tion, n. Want of exact correspondence. [Obs.] --Puller.

Incomplete equation

Incomplete In`com*plete", a. [L. incompletus: cf. F. incomplet. See In- not, and Complete.] 1. Not complete; not filled up; not finished; not having all its parts, or not having them all adjusted; imperfect; defective. A most imperfect and incomplete divine. --Milton. 2. (Bot.) Wanting any of the usual floral organs; -- said of a flower. Incomplete equation (Alg.), an equation some of whose terms are wanting; or one in which the coefficient of some one or more of the powers of the unknown quantity is equal to 0.

Incomplete In`com*plete", a. [L. incompletus: cf. F. incomplet. See In- not, and Complete.] 1. Not complete; not filled up; not finished; not having all its parts, or not having them all adjusted; imperfect; defective. A most imperfect and incomplete divine. --Milton. 2. (Bot.) Wanting any of the usual floral organs; -- said of a flower. Incomplete equation (Alg.), an equation some of whose terms are wanting; or one in which the coefficient of some one or more of the powers of the unknown quantity is equal to 0.

Indeterminate equation

Indeterminate In`de*ter"mi*nate, a. [L. indeterminatus.] Not determinate; not certain or fixed; indefinite; not precise; as, an indeterminate number of years. --Paley. Indeterminate analysis (Math.), that branch of analysis which has for its object the solution of indeterminate problems. Indeterminate coefficients (Math.), coefficients arbitrarily assumed for convenience of calculation, or to facilitate some artifice of analysis. Their values are subsequently determined. Indeterminate equation (Math.), an equation in which the unknown quantities admit of an infinite number of values, or sets of values. A group of equations is indeterminate when it contains more unknown quantities than there are equations. Indeterminate inflorescence (Bot.), a mode of inflorescence in which the flowers all arise from axillary buds, the terminal bud going on to grow and sometimes continuing the stem indefinitely; -- called also acropetal, botryose, centripetal, & indefinite inflorescence. --Gray. Indeterminate problem (Math.), a problem which admits of an infinite number of solutions, or one in which there are fewer imposed conditions than there are unknown or required results. Indeterminate quantity (Math.), a quantity which has no fixed value, but which may be varied in accordance with any proposed condition. Indeterminate series (Math.), a series whose terms proceed by the powers of an indeterminate quantity, sometimes also with indeterminate exponents, or indeterminate coefficients. -- In`de*ter"mi*nate*ly adv. -- In`de*ter"mi*nate*ness, n.

Indeterminate In`de*ter"mi*nate, a. [L. indeterminatus.] Not determinate; not certain or fixed; indefinite; not precise; as, an indeterminate number of years. --Paley. Indeterminate analysis (Math.), that branch of analysis which has for its object the solution of indeterminate problems. Indeterminate coefficients (Math.), coefficients arbitrarily assumed for convenience of calculation, or to facilitate some artifice of analysis. Their values are subsequently determined. Indeterminate equation (Math.), an equation in which the unknown quantities admit of an infinite number of values, or sets of values. A group of equations is indeterminate when it contains more unknown quantities than there are equations. Indeterminate inflorescence (Bot.), a mode of inflorescence in which the flowers all arise from axillary buds, the terminal bud going on to grow and sometimes continuing the stem indefinitely; -- called also acropetal, botryose, centripetal, & indefinite inflorescence. --Gray. Indeterminate problem (Math.), a problem which admits of an infinite number of solutions, or one in which there are fewer imposed conditions than there are unknown or required results. Indeterminate quantity (Math.), a quantity which has no fixed value, but which may be varied in accordance with any proposed condition. Indeterminate series (Math.), a series whose terms proceed by the powers of an indeterminate quantity, sometimes also with indeterminate exponents, or indeterminate coefficients. -- In`de*ter"mi*nate*ly adv. -- In`de*ter"mi*nate*ness, n.

Inequation

Inequation In`e*qua"tion, n. (Math.) An inequality.

Inequation In`e*qua"tion, n. (Math.) An inequality.

Intrinsic equation of a curve

Intrinsic In*trin"sic ([i^]n*tr[i^]n"s[i^]k), a. [L. intrinsecus inward, on the inside; intra within + secus otherwise, beside; akin to E. second: cf. F. intrins[`e]que. See Inter-, Second, and cf. Extrinsic.] 1. Inward; internal; hence, true; genuine; real; essential; inherent; not merely apparent or accidental; -- opposed to extrinsic; as, the intrinsic value of gold or silver; the intrinsic merit of an action; the intrinsic worth or goodness of a person. He was better qualified than they to estimate justly the intrinsic value of Grecian philosophy and refinement. --I. Taylor. 2. (Anat.) Included wholly within an organ or limb, as certain groups of muscles; -- opposed to extrinsic. Intrinsic energy of a body (Physics), the work it can do in virtue of its actual condition, without any supply of energy from without. Intrinsic equation of a curve (Geom.), the equation which expresses the relation which the length of a curve, measured from a given point of it, to a movable point, has to the angle which the tangent to the curve at the movable point makes with a fixed line. Intrinsic value. See the Note under Value, n. Syn: Inherent; innate; natural; real; genuine.

Intrinsic In*trin"sic ([i^]n*tr[i^]n"s[i^]k), a. [L. intrinsecus inward, on the inside; intra within + secus otherwise, beside; akin to E. second: cf. F. intrins[`e]que. See Inter-, Second, and cf. Extrinsic.] 1. Inward; internal; hence, true; genuine; real; essential; inherent; not merely apparent or accidental; -- opposed to extrinsic; as, the intrinsic value of gold or silver; the intrinsic merit of an action; the intrinsic worth or goodness of a person. He was better qualified than they to estimate justly the intrinsic value of Grecian philosophy and refinement. --I. Taylor. 2. (Anat.) Included wholly within an organ or limb, as certain groups of muscles; -- opposed to extrinsic. Intrinsic energy of a body (Physics), the work it can do in virtue of its actual condition, without any supply of energy from without. Intrinsic equation of a curve (Geom.), the equation which expresses the relation which the length of a curve, measured from a given point of it, to a movable point, has to the angle which the tangent to the curve at the movable point makes with a fixed line. Intrinsic value. See the Note under Value, n. Syn: Inherent; innate; natural; real; genuine.

Personal equation

Personal Per"son*al, a. [L. personalis: cf. F. personnel.] 1. Pertaining to human beings as distinct from things. Every man so termed by way of personal difference. --Hooker. 2. Of or pertaining to a particular person; relating to, or affecting, an individual, or each of many individuals; peculiar or proper to private concerns; not public or general; as, personal comfort; personal desire. The words are conditional, -- If thou doest well, -- and so personal to Cain. --Locke. 3. Pertaining to the external or bodily appearance; corporeal; as, personal charms. --Addison. 4. Done in person; without the intervention of another. ``Personal communication.' --Fabyan. The immediate and personal speaking of God. --White. 5. Relating to an individual, his character, conduct, motives, or private affairs, in an invidious and offensive manner; as, personal reflections or remarks. 6. (Gram.) Denoting person; as, a personal pronoun. Personal action (Law), a suit or action by which a man claims a debt or personal duty, or damages in lieu of it; or wherein he claims satisfaction in damages for an injury to his person or property, or the specific recovery of goods or chattels; -- opposed to real action. Personal equation. (Astron.) See under Equation. Personal estate or property (Law), movables; chattels; -- opposed to real estate or property. It usually consists of things temporary and movable, including all subjects of property not of a freehold nature. Personal identity (Metaph.), the persistent and continuous unity of the individual person, which is attested by consciousness. Personal pronoun (Gram.), one of the pronouns I, thou, he, she, it, and their plurals. Personal representatives (Law), the executors or administrators of a person deceased. Personal rights, rights appertaining to the person; as, the rights of a personal security, personal liberty, and private property. Personal tithes. See under Tithe. Personal verb (Gram.), a verb which is modified or inflected to correspond with the three persons.

Personal Per"son*al, a. [L. personalis: cf. F. personnel.] 1. Pertaining to human beings as distinct from things. Every man so termed by way of personal difference. --Hooker. 2. Of or pertaining to a particular person; relating to, or affecting, an individual, or each of many individuals; peculiar or proper to private concerns; not public or general; as, personal comfort; personal desire. The words are conditional, -- If thou doest well, -- and so personal to Cain. --Locke. 3. Pertaining to the external or bodily appearance; corporeal; as, personal charms. --Addison. 4. Done in person; without the intervention of another. ``Personal communication.' --Fabyan. The immediate and personal speaking of God. --White. 5. Relating to an individual, his character, conduct, motives, or private affairs, in an invidious and offensive manner; as, personal reflections or remarks. 6. (Gram.) Denoting person; as, a personal pronoun. Personal action (Law), a suit or action by which a man claims a debt or personal duty, or damages in lieu of it; or wherein he claims satisfaction in damages for an injury to his person or property, or the specific recovery of goods or chattels; -- opposed to real action. Personal equation. (Astron.) See under Equation. Personal estate or property (Law), movables; chattels; -- opposed to real estate or property. It usually consists of things temporary and movable, including all subjects of property not of a freehold nature. Personal identity (Metaph.), the persistent and continuous unity of the individual person, which is attested by consciousness. Personal pronoun (Gram.), one of the pronouns I, thou, he, she, it, and their plurals. Personal representatives (Law), the executors or administrators of a person deceased. Personal rights, rights appertaining to the person; as, the rights of a personal security, personal liberty, and private property. Personal tithes. See under Tithe. Personal verb (Gram.), a verb which is modified or inflected to correspond with the three persons.

Reciprocal equation

Reciprocal Re*cip"ro*cal, a. [L. reciprocus; of unknown origin.] 1. Recurring in vicissitude; alternate. 2. Done by each to the other; interchanging or interchanged; given and received; due from each to each; mutual; as, reciprocal love; reciprocal duties. Let our reciprocal vows be remembered. --Shak. 3. Mutually interchangeable. These two rules will render a definition reciprocal with the thing defined. --I. Watts. 4. (Gram.) Reflexive; -- applied to pronouns and verbs, but sometimes limited to such pronouns as express mutual action. 5. (Math.) Used to denote different kinds of mutual relation; often with reference to the substitution of reciprocals for given quantities. See the Phrases below. Reciprocal equation (Math.), one which remains unchanged in form when the reciprocal of the unknown quantity is substituted for that quantity. Reciprocal figures (Geom.), two figures of the same kind (as triangles, parallelograms, prisms, etc.), so related that two sides of the one form the extremes of a proportion of which the means are the two corresponding sides of the other; in general, two figures so related that the first corresponds in some special way to the second, and the second corresponds in the same way to the first. Reciprocal proportion (Math.), a proportion such that, of four terms taken in order, the first has to the second the same ratio which the fourth has to the third, or the first has to the second the same ratio which the reciprocal of the third has to the reciprocal of the fourth. Thus, 2:5: :20:8 form a reciprocal proportion, because 2:5: :1/20:1/8. Reciprocal quantities (Math.), any two quantities which produce unity when multiplied together. Reciprocal ratio (Math.), the ratio between the reciprocals of two quantities; as, the reciprocal ratio of 4 to 9 is that of 3/4 to 1/9. Reciprocal terms (Logic), those terms which have the same signification, and, consequently, are convertible, and may be used for each other. Syn: Mutual; alternate. Usage: Reciprocal, Mutual. The distinctive idea of mutual is, that the parties unite by interchange in the same act; as, a mutual covenant; mutual affection, etc. The distinctive idea of reciprocal is, that one party acts by way of return or response to something previously done by the other party; as, a reciprocal kindness; reciprocal reproaches, etc. Love is reciprocal when the previous affection of one party has drawn forth the attachment of the other. To make it mutual in the strictest sense, the two parties should have fallen in love at the same time; but as the result is the same, the two words are here used interchangeably. The ebbing and flowing of the tide is a case where the action is reciprocal, but not mutual.

Reciprocal Re*cip"ro*cal, a. [L. reciprocus; of unknown origin.] 1. Recurring in vicissitude; alternate. 2. Done by each to the other; interchanging or interchanged; given and received; due from each to each; mutual; as, reciprocal love; reciprocal duties. Let our reciprocal vows be remembered. --Shak. 3. Mutually interchangeable. These two rules will render a definition reciprocal with the thing defined. --I. Watts. 4. (Gram.) Reflexive; -- applied to pronouns and verbs, but sometimes limited to such pronouns as express mutual action. 5. (Math.) Used to denote different kinds of mutual relation; often with reference to the substitution of reciprocals for given quantities. See the Phrases below. Reciprocal equation (Math.), one which remains unchanged in form when the reciprocal of the unknown quantity is substituted for that quantity. Reciprocal figures (Geom.), two figures of the same kind (as triangles, parallelograms, prisms, etc.), so related that two sides of the one form the extremes of a proportion of which the means are the two corresponding sides of the other; in general, two figures so related that the first corresponds in some special way to the second, and the second corresponds in the same way to the first. Reciprocal proportion (Math.), a proportion such that, of four terms taken in order, the first has to the second the same ratio which the fourth has to the third, or the first has to the second the same ratio which the reciprocal of the third has to the reciprocal of the fourth. Thus, 2:5: :20:8 form a reciprocal proportion, because 2:5: :1/20:1/8. Reciprocal quantities (Math.), any two quantities which produce unity when multiplied together. Reciprocal ratio (Math.), the ratio between the reciprocals of two quantities; as, the reciprocal ratio of 4 to 9 is that of 3/4 to 1/9. Reciprocal terms (Logic), those terms which have the same signification, and, consequently, are convertible, and may be used for each other. Syn: Mutual; alternate. Usage: Reciprocal, Mutual. The distinctive idea of mutual is, that the parties unite by interchange in the same act; as, a mutual covenant; mutual affection, etc. The distinctive idea of reciprocal is, that one party acts by way of return or response to something previously done by the other party; as, a reciprocal kindness; reciprocal reproaches, etc. Love is reciprocal when the previous affection of one party has drawn forth the attachment of the other. To make it mutual in the strictest sense, the two parties should have fallen in love at the same time; but as the result is the same, the two words are here used interchangeably. The ebbing and flowing of the tide is a case where the action is reciprocal, but not mutual.

Root of an equation

2. An edible or esculent root, especially of such plants as produce a single root, as the beet, carrot, etc.; as, the root crop. 3. That which resembles a root in position or function, esp. as a source of nourishment or support; that from which anything proceeds as if by growth or development; as, the root of a tooth, a nail, a cancer, and the like. Specifically: (a) An ancestor or progenitor; and hence, an early race; a stem. They were the roots out of which sprang two distinct people. --Locke. (b) A primitive form of speech; one of the earliest terms employed in language; a word from which other words are formed; a radix, or radical. (c) The cause or occasion by which anything is brought about; the source. ``She herself . . . is root of bounty.' --Chaucer. The love of money is a root of all kinds of evil. --1 Tim. vi. 10 (rev. Ver.) (d) (Math.) That factor of a quantity which when multiplied into itself will produce that quantity; thus, 3 is a root of 9, because 3 multiplied into itself produces 9; 3 is the cube root of 27. (e) (Mus.) The fundamental tone of any chord; the tone from whose harmonics, or overtones, a chord is composed. --Busby. (f) The lowest place, position, or part. ``Deep to the roots of hell.' --Milton. ``The roots of the mountains.' --Southey. 4. (Astrol.) The time which to reckon in making calculations. When a root is of a birth yknowe [known]. --Chaucer. A["e]rial roots. (Bot.) (a) Small roots emitted from the stem of a plant in the open air, which, attaching themselves to the bark of trees, etc., serve to support the plant. (b) Large roots growing from the stem, etc., which descend and establish themselves in the soil. See Illust. of Mangrove. Multiple primary root (Bot.), a name given to the numerous roots emitted from the radicle in many plants, as the squash. Primary root (Bot.), the central, first-formed, main root, from which the rootlets are given off. Root and branch, every part; wholly; completely; as, to destroy an error root and branch. Root-and-branch men, radical reformers; -- a designation applied to the English Independents (1641). See Citation under Radical, n., 2. Root barnacle (Zo["o]l.), one of the Rhizocephala. Root hair (Bot.), one of the slender, hairlike fibers found on the surface of fresh roots. They are prolongations of the superficial cells of the root into minute tubes. --Gray. Root leaf (Bot.), a radical leaf. See Radical, a., 3 (b) . Root louse (Zo["o]l.), any plant louse, or aphid, which lives on the roots of plants, as the Phylloxera of the grapevine. See Phylloxera. Root of an equation (Alg.), that value which, substituted for the unknown quantity in an equation, satisfies the equation. Root of a nail (Anat.), the part of a nail which is covered by the skin. Root of a tooth (Anat.), the part of a tooth contained in the socket and consisting of one or more fangs. Secondary roots (Bot.), roots emitted from any part of the plant above the radicle. To strike root, To take root, to send forth roots; to become fixed in the earth, etc., by a root; hence, in general, to become planted, fixed, or established; to increase and spread; as, an opinion takes root. ``The bended twigs take root.' --Milton.

2. An edible or esculent root, especially of such plants as produce a single root, as the beet, carrot, etc.; as, the root crop. 3. That which resembles a root in position or function, esp. as a source of nourishment or support; that from which anything proceeds as if by growth or development; as, the root of a tooth, a nail, a cancer, and the like. Specifically: (a) An ancestor or progenitor; and hence, an early race; a stem. They were the roots out of which sprang two distinct people. --Locke. (b) A primitive form of speech; one of the earliest terms employed in language; a word from which other words are formed; a radix, or radical. (c) The cause or occasion by which anything is brought about; the source. ``She herself . . . is root of bounty.' --Chaucer. The love of money is a root of all kinds of evil. --1 Tim. vi. 10 (rev. Ver.) (d) (Math.) That factor of a quantity which when multiplied into itself will produce that quantity; thus, 3 is a root of 9, because 3 multiplied into itself produces 9; 3 is the cube root of 27. (e) (Mus.) The fundamental tone of any chord; the tone from whose harmonics, or overtones, a chord is composed. --Busby. (f) The lowest place, position, or part. ``Deep to the roots of hell.' --Milton. ``The roots of the mountains.' --Southey. 4. (Astrol.) The time which to reckon in making calculations. When a root is of a birth yknowe [known]. --Chaucer. A["e]rial roots. (Bot.) (a) Small roots emitted from the stem of a plant in the open air, which, attaching themselves to the bark of trees, etc., serve to support the plant. (b) Large roots growing from the stem, etc., which descend and establish themselves in the soil. See Illust. of Mangrove. Multiple primary root (Bot.), a name given to the numerous roots emitted from the radicle in many plants, as the squash. Primary root (Bot.), the central, first-formed, main root, from which the rootlets are given off. Root and branch, every part; wholly; completely; as, to destroy an error root and branch. Root-and-branch men, radical reformers; -- a designation applied to the English Independents (1641). See Citation under Radical, n., 2. Root barnacle (Zo["o]l.), one of the Rhizocephala. Root hair (Bot.), one of the slender, hairlike fibers found on the surface of fresh roots. They are prolongations of the superficial cells of the root into minute tubes. --Gray. Root leaf (Bot.), a radical leaf. See Radical, a., 3 (b) . Root louse (Zo["o]l.), any plant louse, or aphid, which lives on the roots of plants, as the Phylloxera of the grapevine. See Phylloxera. Root of an equation (Alg.), that value which, substituted for the unknown quantity in an equation, satisfies the equation. Root of a nail (Anat.), the part of a nail which is covered by the skin. Root of a tooth (Anat.), the part of a tooth contained in the socket and consisting of one or more fangs. Secondary roots (Bot.), roots emitted from any part of the plant above the radicle. To strike root, To take root, to send forth roots; to become fixed in the earth, etc., by a root; hence, in general, to become planted, fixed, or established; to increase and spread; as, an opinion takes root. ``The bended twigs take root.' --Milton.

Simple equation

12. (Min.) Homogenous. 13. (Zo["o]l.) Consisting of a single individual or zooid; as, a simple ascidian; -- opposed to compound. Simple contract (Law), any contract, whether verbal or written, which is not of record or under seal. --J. W. Smith. --Chitty. Simple equation (Alg.), an eqyation containing but one unknown quantity, and that quantity only in the first degree. Simple eye (Zo["o]l.), an eye having a single lens; -- opposed to compound eye. Simple interest. See under Interest. Simple larceny. (Law) See under Larceny. Simple obligation (Rom. Law), an obligation which does not depend for its execution upon any event provided for by the parties, or is not to become void on the happening of any such event. --Burrill. Syn: Single; uncompounded; unmingled; unmixed; mere; uncombined; elementary; plain; artless; sincere; harmless; undesigning; frank; open; unaffected; inartificial; unadorned; credulous; silly; foolish; shallow; unwise. Usage: Simple, Silly. One who is simple is sincere, unaffected, and inexperienced in duplicity, -- hence liable to be duped. A silly person is one who is ignorant or weak and also self-confident; hence, one who shows in speech and act a lack of good sense. Simplicity is incompatible with duplicity, artfulness, or vanity, while silliness is consistent with all three. Simplicity denotes lack of knowledge or of guile; silliness denotes want of judgment or right purpose, a defect of character as well as of education. I am a simple woman, much too weak To oppose your cunning. --Shak. He is the companion of the silliest people in their most silly pleasure; he is ready for every impertinent entertainment and diversion. --Law.

12. (Min.) Homogenous. 13. (Zo["o]l.) Consisting of a single individual or zooid; as, a simple ascidian; -- opposed to compound. Simple contract (Law), any contract, whether verbal or written, which is not of record or under seal. --J. W. Smith. --Chitty. Simple equation (Alg.), an eqyation containing but one unknown quantity, and that quantity only in the first degree. Simple eye (Zo["o]l.), an eye having a single lens; -- opposed to compound eye. Simple interest. See under Interest. Simple larceny. (Law) See under Larceny. Simple obligation (Rom. Law), an obligation which does not depend for its execution upon any event provided for by the parties, or is not to become void on the happening of any such event. --Burrill. Syn: Single; uncompounded; unmingled; unmixed; mere; uncombined; elementary; plain; artless; sincere; harmless; undesigning; frank; open; unaffected; inartificial; unadorned; credulous; silly; foolish; shallow; unwise. Usage: Simple, Silly. One who is simple is sincere, unaffected, and inexperienced in duplicity, -- hence liable to be duped. A silly person is one who is ignorant or weak and also self-confident; hence, one who shows in speech and act a lack of good sense. Simplicity is incompatible with duplicity, artfulness, or vanity, while silliness is consistent with all three. Simplicity denotes lack of knowledge or of guile; silliness denotes want of judgment or right purpose, a defect of character as well as of education. I am a simple woman, much too weak To oppose your cunning. --Shak. He is the companion of the silliest people in their most silly pleasure; he is ready for every impertinent entertainment and diversion. --Law.

Simultaneous equations

Simultaneous Si`mul*ta"ne*ous, a. [LL. simultim at the same time, fr. L. simul. See Simulate.] Existing, happening, or done, at the same time; as, simultaneous events. -- Si`mul*ta"ne*ous*ly, adv. -- Si`mul*ta"ne*ous*ness, n. Simultaneous equations (Alg.), two or more equations in which the values of the unknown quantities entering them are the same at the same time in both or in all.

Simultaneous Si`mul*ta"ne*ous, a. [LL. simultim at the same time, fr. L. simul. See Simulate.] Existing, happening, or done, at the same time; as, simultaneous events. -- Si`mul*ta"ne*ous*ly, adv. -- Si`mul*ta"ne*ous*ness, n. Simultaneous equations (Alg.), two or more equations in which the values of the unknown quantities entering them are the same at the same time in both or in all.

To reduce an equation

4. To bring to a certain state or condition by grinding, pounding, kneading, rubbing, etc.; as, to reduce a substance to powder, or to a pasty mass; to reduce fruit, wood, or paper rags, to pulp. It were but right And equal to reduce me to my dust. --Milton. 5. To bring into a certain order, arrangement, classification, etc.; to bring under rules or within certain limits of descriptions and terms adapted to use in computation; as, to reduce animals or vegetables to a class or classes; to reduce a series of observations in astronomy; to reduce language to rules. 6. (Arith.) (a) To change, as numbers, from one denomination into another without altering their value, or from one denomination into others of the same value; as, to reduce pounds, shillings, and pence to pence, or to reduce pence to pounds; to reduce days and hours to minutes, or minutes to days and hours. (b) To change the form of a quantity or expression without altering its value; as, to reduce fractions to their lowest terms, to a common denominator, etc. 7. (Chem.) To bring to the metallic state by separating from impurities; hence, in general, to remove oxygen from; to deoxidize; to combine with, or to subject to the action of, hydrogen; as, ferric iron is reduced to ferrous iron; or metals are reduced from their ores; -- opposed to oxidize. 8. (Med.) To restore to its proper place or condition, as a displaced organ or part; as, to reduce a dislocation, a fracture, or a hernia. Reduced iron (Chem.), metallic iron obtained through deoxidation of an oxide of iron by exposure to a current of hydrogen or other reducing agent. When hydrogen is used the product is called also iron by hydrogen. To reduce an equation (Alg.), to bring the unknown quantity by itself on one side, and all the known quantities on the other side, without destroying the equation. To reduce an expression (Alg.), to obtain an equivalent expression of simpler form. To reduce a square (Mil.), to reform the line or column from the square. Syn: To diminish; lessen; decrease; abate; shorten; curtail; impair; lower; subject; subdue; subjugate; conquer.

4. To bring to a certain state or condition by grinding, pounding, kneading, rubbing, etc.; as, to reduce a substance to powder, or to a pasty mass; to reduce fruit, wood, or paper rags, to pulp. It were but right And equal to reduce me to my dust. --Milton. 5. To bring into a certain order, arrangement, classification, etc.; to bring under rules or within certain limits of descriptions and terms adapted to use in computation; as, to reduce animals or vegetables to a class or classes; to reduce a series of observations in astronomy; to reduce language to rules. 6. (Arith.) (a) To change, as numbers, from one denomination into another without altering their value, or from one denomination into others of the same value; as, to reduce pounds, shillings, and pence to pence, or to reduce pence to pounds; to reduce days and hours to minutes, or minutes to days and hours. (b) To change the form of a quantity or expression without altering its value; as, to reduce fractions to their lowest terms, to a common denominator, etc. 7. (Chem.) To bring to the metallic state by separating from impurities; hence, in general, to remove oxygen from; to deoxidize; to combine with, or to subject to the action of, hydrogen; as, ferric iron is reduced to ferrous iron; or metals are reduced from their ores; -- opposed to oxidize. 8. (Med.) To restore to its proper place or condition, as a displaced organ or part; as, to reduce a dislocation, a fracture, or a hernia. Reduced iron (Chem.), metallic iron obtained through deoxidation of an oxide of iron by exposure to a current of hydrogen or other reducing agent. When hydrogen is used the product is called also iron by hydrogen. To reduce an equation (Alg.), to bring the unknown quantity by itself on one side, and all the known quantities on the other side, without destroying the equation. To reduce an expression (Alg.), to obtain an equivalent expression of simpler form. To reduce a square (Mil.), to reform the line or column from the square. Syn: To diminish; lessen; decrease; abate; shorten; curtail; impair; lower; subject; subdue; subjugate; conquer.

Transcendental equation

Trancscendental Tranc`scen*den"tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. 2. (Philos.) In the Kantian system, of or pertaining to that which can be determined a priori in regard to the fundamental principles of all human knowledge. What is transcendental, therefore, transcends empiricism; but is does not transcend all human knowledge, or become transcendent. It simply signifies the a priori or necessary conditions of experience which, though affording the conditions of experience, transcend the sphere of that contingent knowledge which is acquired by experience. 3. Vaguely and ambitiously extravagant in speculation, imagery, or diction. Note: In mathematics, a quantity is said to be transcendental relative to another quantity when it is expressed as a transcendental function of the latter; thus, a^x, 10^2x, log x, sin x, tan x, etc., are transcendental relative to x. Transcendental curve (Math.), a curve in which one ordinate is a transcendental function of the other. Transcendental equation (Math.), an equation into which a transcendental function of one of the unknown or variable quantities enters. Transcendental function. (Math.) See under Function. Syn: Transcendental, Empirical. Usage: These terms, with the corresponding nouns, transcendentalism and empiricism, are of comparatively recent origin. Empirical refers to knowledge which is gained by the experience of actual phenomena, without reference to the principles or laws to which they are to be referred, or by which they are to be explained. Transcendental has reference to those beliefs or principles which are not derived from experience, and yet are absolutely necessary to make experience possible or useful. Such, in the better sense of the term, is the transcendental philosophy, or transcendentalism. Each of these words is also used in a bad sense, empiricism applying to that one-sided view of knowledge which neglects or loses sight of the truths or principles referred to above, and trusts to experience alone; transcendentalism, to the opposite extreme, which, in its deprecation of experience, loses sight of the relations which facts and phenomena sustain to principles, and hence to a kind of philosophy, or a use of language, which is vague, obscure, fantastic, or extravagant.

Trancscendental Tranc`scen*den"tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. 2. (Philos.) In the Kantian system, of or pertaining to that which can be determined a priori in regard to the fundamental principles of all human knowledge. What is transcendental, therefore, transcends empiricism; but is does not transcend all human knowledge, or become transcendent. It simply signifies the a priori or necessary conditions of experience which, though affording the conditions of experience, transcend the sphere of that contingent knowledge which is acquired by experience. 3. Vaguely and ambitiously extravagant in speculation, imagery, or diction. Note: In mathematics, a quantity is said to be transcendental relative to another quantity when it is expressed as a transcendental function of the latter; thus, a^x, 10^2x, log x, sin x, tan x, etc., are transcendental relative to x. Transcendental curve (Math.), a curve in which one ordinate is a transcendental function of the other. Transcendental equation (Math.), an equation into which a transcendental function of one of the unknown or variable quantities enters. Transcendental function. (Math.) See under Function. Syn: Transcendental, Empirical. Usage: These terms, with the corresponding nouns, transcendentalism and empiricism, are of comparatively recent origin. Empirical refers to knowledge which is gained by the experience of actual phenomena, without reference to the principles or laws to which they are to be referred, or by which they are to be explained. Transcendental has reference to those beliefs or principles which are not derived from experience, and yet are absolutely necessary to make experience possible or useful. Such, in the better sense of the term, is the transcendental philosophy, or transcendentalism. Each of these words is also used in a bad sense, empiricism applying to that one-sided view of knowledge which neglects or loses sight of the truths or principles referred to above, and trusts to experience alone; transcendentalism, to the opposite extreme, which, in its deprecation of experience, loses sight of the relations which facts and phenomena sustain to principles, and hence to a kind of philosophy, or a use of language, which is vague, obscure, fantastic, or extravagant.

Zonal equation

Zonal on"al, a. [L. zonalis.] Of or pertaining to a zone; having the form of a zone or zones. Zonal equation (Crystallog.), the mathematical relation which belongs to all the planes of a zone, and expresses their common position with reference to the axes. Zonal structure (Crystallog.), a structure characterized by the arrangements of color, inclusions, etc., of a crystal in parallel or concentric layers, which usually follow the outline of the crystal, and mark the changes that have taken place during its growth. Zonal symmetry. (Biol.) See the Note under Symmetry.

Zonal on"al, a. [L. zonalis.] Of or pertaining to a zone; having the form of a zone or zones. Zonal equation (Crystallog.), the mathematical relation which belongs to all the planes of a zone, and expresses their common position with reference to the axes. Zonal structure (Crystallog.), a structure characterized by the arrangements of color, inclusions, etc., of a crystal in parallel or concentric layers, which usually follow the outline of the crystal, and mark the changes that have taken place during its growth. Zonal symmetry. (Biol.) See the Note under Symmetry.

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- In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions...

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- mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions...

- In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0...

- In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable...

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