reductt (redukt)

t. [ reduce. All the kynges host there beying a embled and reducte into one companye. Hall, Edw. IV., an. 10. Pray let me reduct some two or three shillings for points and ribands. B. Jonson , Every Man out of his Humour, iv. 5. reduct (re-dukt'), n. [ reductus , a withdrawing-place: see redout2 .] In building , a lit D A drawing a figure whose parts are supposed to have certain relations, and then showing that this leads to a conclusion contrary to a known proposition, whence it follows that the parts of the figure cannot have those relations. Thus, in Euclid's "Elements" the proposition that if a triangle has two angles equal the sides opposite those angles will be equal is proved as follows. In the triangle ABC, let the angles ABC and ACB be equal. Then, suppose AB to be greater than AC. Lay off BD = AC and join DC. Then, comparing the two triangles ACB and DBC, we have in the former the sides AC and BC and their included angle ACB equal in the latter to the sides DB and CB and their included angle DBC. Hence, these two triangles would be equal, or the part would be equal to the whole. This proof is a reductio ad absurdum. This kind of reasoning is considered somewhat objectionable as not showing the principle from which the proposi- B tion flows; but it is a perfectly conclusive mode of proof, and, in fact, is in all cases readily converted into a direct proof. Thus, in the above example, we have only to compare the triangle ABC with itself, considering it as two triangles according as the angle B is named before C or vice vers a. In the triangle ABC the angles Band C with the included side BC are respectively equal in the triangle ACB to the angles C and B with the included side CB; hence the other parts of the triangles are equal, and the side AC opposite the first angle Bin the first triangle is equal to the side AB opposite the first angle C in the second triangle. reduction (re-duk's hon), n. C [ reduction , F. réduction = Pr. reductio = Sp. reduccion = Pg. reducção = It. riduzione, For reduction of your majesty's realm of Ireland to the unity of the Church. Bp. Burnet , Records, II. ii. ( b ) Conversion into another state or form: as, the reduc tion of a body to powder; the reduction of things to order. (c) Diminution: as, the reduction of the expenses of government; the reduction of the national debt; a reduc tion of 25 per cent. made to wholesale buyers. Let him therefore first make the proper reduction in the account, and then see what it amounts to. Water land , Works, VI. 186. (d) Conquest; subjugation: as, the reduction of a province under the power of a foreign nation; the reduction of a fortre . (e) A settlement or parish of South American Indians converted and trained by the Jesuits. Governing and civilizing the natives of Brazil and Paraguay in the mi ions and reductions , or ministering, at the hourly risk of his life, to his coreligionists in England under Elizabeth and James I., the Jesuit appears alike devoted, indefatigable, cheerful, and worthy of hearty admiration and respect. Encyc . Brit ., XIII . 649. The Indians [under the Jesuits in Paraguay] were gath ered into towns or communal villages called bourgaden or reductions , where they were taught the common arts, agriculture, and the practice of rearing cattle. Johns Hopkins Univ . Studies , 8th ser., IV. 32. (f) The bringing of a problem to depend on a problem already solved. (g) The transformation of an algebraic expre ion into another of a simpler kind. (h) The low ering of the values of the numerator and denominator of a fraction, or of the antecedent and consequent of a ratio, by dividing both by the same quantity. (i) The conversion of a quantity expre ed in terms of one denomination so as to expre it in terms of another denomination. Ascending reduction is conversion to terms of larger units; descending reduction, conversion to terms of smaller units. ()) The proof of the conclusion of an indirect syllogism from its premises by means of a direct syllogism and immediate inferences. This is said to be a reduction to the mode of direct syllogism employed. ( k ) A direct syllogism proving, by means of conversions and other immediate inferences, that the conclusion of an indirect syllogism follows from its premises. (1) The act or proce of making a copy of a figure, map, design, draft, etc., on a smaller scale, preserving the original proportions; also, the result of this proce . (m) In surg., the operation of restoring a dislocated or fractured bone to its former place. (n) Separation of a metal from substances combined with it: used especially with reference to lead, zinc, and copper, and also applied to the treatment of iron ore, as when steel is made from it by a direct proce . (o) In astron., the correction of observed quantities for instrumental errors, as well as for refraction, parallax, aberration, prece ion, and nutation, so as to bring out their cosmical significance. A similar proce is applied to observations in other physical sciences. (p) In Scots law, an action for setting aside a deed, writing, etc.Apagogical reduction, in logic, a reduction in which the contradictory of the conclusion becomes one of the premises, and the contradictory of one of the premises the conclusion. Apagogical reduction is an application of the reductio ad absurdum, and is also called reductio per impo ibile. Example: IMG:content-1803.png:[blocks in formation] Chasles-Zeuthen reduction, a method of finding how many figures fulfil certain conditions, by the consideration of degenerate figures composed of simpler figures with lower constants. Thus, in this way we readily find that the number of conics touching five given conics in a plane is 3,264.- Iron-reduction proce . See pro ce . Long reduction, in logic, a reduction in which the major premise of the original syllogism becomes the minor premise, and vice vers a, and in which one of the premises and the conclusion are converted. Example: IMG:content-1804.png:[blocks in formation] Ostensive reduction, that reduction which has for its premises the original premises or their conversions, and for its conclusion the original conclusion or its converse. -Reduction and reduction-improbation, in Scots law, the designations given to the two varieties of resci ory actions. See improbation, 2. - Reduction reductive, an action in which a decree of reduction which has been erroneously or improperly obtained is sought to be reduced. -Reduction to the ecliptic, the difference between the anomaly of a planet reckoned from its node and the longitude reckoned from the same point. - Short reduction, in logic, a reduction which differs from the original syllogism only in having one of its premises converted. The following is an example: IMG:content-1805.png:[blocks in formation] = Syn. (c) Le ening, decrease, abatement, curtailment, abridgment, contraction, retrenchment. reduction-compa es (re-duk's hon-kum"pasez), n. pl. Proportional dividers, or whole-and half dividers. reduction-formula (rē-duk's hon-fôr mū-lä), п. In the integral calculus, a formula depending on integration by parts, reducing an integral to another nearer to one of the standard forms. reduction-works (re-duk's hon-wėrks), n. sing. and pl. A metallurgical establishment; smelting-works. reductive (re-duktiv), a. and n. [= F. réduc tif Sp. Pg. reductivo = It. riduttivo, 〈L. re duct us, pp. of reducere, lead or bring back: see reduct, reduce.] I. a. Having the property, power, or effect of reducing; tending to reduce. Inquire into the repentance of thy former life particularly; whether it were of a great and perfect grief, and productive of fixed resolutions of holy living, and reduc tive of these to act. Jer. Taylor , Holy Dying, iv. 6. Reduction reductive. See reduction. - Reductive conversion, in logic, a conversion of a proposition in which there is some modification of the subject or predicate: as, no man is a mother, therefore no mother is some man. See conversion, 2. - Reductive principle, a principle by which an indirect syllogism is reduced to a direct mood. The reductive principles were said to be conversion, transposition, and reductio per impo ibile. II. n . That which has the power of reducing. So that it should seem there needed no other reductive of the numbers of men to an equability than the wars that have happened in the world. Sir M. Hale , Orig. of Mankind, p. 215. reductively (rē-duktiv-li), adv . By reduction; by consequence. Love, and simplicity, and humility, and usefulne :... I think these do reductively contain all that is excellent in the whole conjugation of Christian graces. Jer. Taylor , Works (ed. 1835), II. 44.