11 E D 



RED 



whofe roots arc — 2, and — 5 ; therefore, the four roots 

 of the propofed equation arc 2, 2,-2, — 5- 



If the roots of -.ai equation be equal, but with contrary 

 figns, the operation is more fimple. For, in this cafe, we 

 have only to change the figns of the roots in the propofed 

 equation, by changing the alternate ilgns of its co-efficient ; 

 and we fhall thus have two equations, having neceffarily 

 a common quadratic faftor of the form x^ — a, which 

 may be found, and the equation depreffed as before. 



Exam. 3. — Required the roots of the equation k' + 3.1'' 

 — 7 A-' — 27 A- — 18 = 0, two of which are equal, but with 

 contrary figns. 



By changing the figns of the alternate terms, we have 

 *•' ~ 3 •■«' ~ 7 x^ + 27 -i- — 18 = 0, the common quadratic 

 divifor of which is a' - 9 = O ; whence the equal roots 

 with contrary figns are + 3, and — 3. Now dividing the 

 propofed equation by x" — 9, we have .\' -|- 3 x -|- 2 =: o ; 



-3 /' 



whence x = — -\- \/ ~ - = — 2, and — i, which are 

 2 — V ^ 



the other two roots. 



On this fubjeft the reader fliould confult Waring's Mcdi- 

 tationes Algebraicse, cap. 3. See alfo Bonnycaille's Al- 

 gebra, vols. i. and li. 



Rf.duction of I liter eft of the public Debt. See Fund. 

 Reduction o/" Carivj. See CiinvE. 

 Reduction of a Figure, Defign, or Draught, is the 

 making a copy of it, either larger or fmaller than the ori- 

 ginal , ftill preferving the form and proportion. 



The great ufe of the proportional compaffes is in the re- 

 duction of figures, &c. whence they are alfo called conj^iT^j 

 of reduBion. 



There are various methods of reducing figures, Sec. ; the 

 moft eafy is by means oi the pentngrnph (which fee), or pa- 

 rallelogram : but this has its defcfts. The bell and moft 

 ufual methods of reduction arc as follow : 



To reduce a figure, as A B C D E [Plate XII. Geometry, 

 fg. 5.) into a lefs compafs. — About tlie middle of the figure, 

 ns ■z., pitch on a point ; and from this point draw lines to its 

 feveral angles, A, B, C, &c. then drawing the line ab pa- 

 rallel to A B, be parallel to B C, &c. you will have the 

 figure ahc de fimilar to A B C D E. 



If the figure abcde had been required to be enlarged, 

 there needed nothing but to produce the lines from the point 

 beyond the angles, as z D, z C, &c. and to draw lines, 

 TOz. DC, C B", &c. parallel to the fides fl?r, cb, &c. 



To reduce a fgure by the angle of proportion. — Suppofe the 

 figure ABCDE {fg- 6.) required to be diminifhed in 

 the proportion of the line A B to iji {fis- ?•) • draw the 

 indefinite line GH {fg. 8.), and, from G to H, fet off 

 the line A B ; on G dilcribe the arc HI; fet off the line 

 a 3 as a chord on HI, and draw G I. Then with the 

 angle I G H you have all the mcafures of the figure to be 

 drawn. Thus to lay down the point c, take the interval 

 B C, and upon the point G defcribe the arc K L ; alfo on 

 the point G defcribe M N ; and upon a, with the diftance 

 M N, defcribe an arc cutting the preceding one in c, which 

 will determine the fide be. And after the fame manner are 

 all the other fides and angles to be defcribed. The fame 

 procefs will alfo ferve to enlarge the figure. 



To reduce a fgure by a fcale. — Meafure all the fides of the 

 figure, e. gr. ABCDE, by a fcale, and lay down the 

 fame meafures refpeftively, from a fmaller fcale in the pro- 

 portion required. 



To reduce a map, defign, or figure, by fquara. — Divide the 

 original into little fquarcs, and divide a frefh paper of the 

 dimenfions required, into the fame number of fquares ; 



which arc to be larger or lefs than the former, as the map is 

 to be enlarged or diminiftied. 



This done, in every fquare of the fccond figure, draw 

 what you find in its correfpondcnt one in the firft. 



Reduction to the Ecliptic, in Aftronomy, is the difference 

 between the argument of latitude, as N P [Plate XIX. 

 Aflronomy, fig. 10.) and an arc of the ecliptic N R, inter- 

 cepted between the place of a planet and the node N. 



To find the reduHion. — The angle of inclination PNR, 

 and the argument of latitude N P, being given, find, by the 

 dodlrine of fpherics, the arc N R ; fubtracl N R and N P 

 from each other, the remainder is the reduftion» 



Reduction into firfl Matter, is a term which alchemifts 

 formerly ufed, when they found their fubftances putrefy, 

 and grow black. 



Reduction was more particularly ufed for the conver- 

 fion of a dry matter into a liquid, particularly into water ; 

 which by the akhemifts was held the principle of al) 

 things. 



The rcduftion of metals into their firft matter, or prin- 

 ciples, according to thefe philofophers, can only be effefted 

 by mercury ; nothing elfe being able to loofen the fixed ful- 

 phur of metallic bodies, which binds them together. 



Reduction, in Metallurgy, is the decompofition of a 

 metallic oxyd, fo as to leave tlie metal in a ftate of greater 

 or lefs purity, and exhibiting the lultre which is fo emi- 

 nently charafteriltic of metallic bodies. 



Reduction is, for the moft part, effefted by charcoal and 

 a high temperature, either with or without the affiftance of 

 fluxes. For the various modes of reduftion in a£lual prac- 

 tice, fee the articles of the different metals. 



Reduction, in Surgery, denotes an operation by which 

 a diflocated, luxated, or fraftured bone is reftored to its 

 former place. Rcduftion, or repofition, is always to be per- 

 formed before any remedy be apphed. 



REDUIT, in Military Affairs. See Reduct. 



REDUNDANCY, or Redundance, a fault in dif- 

 courfe, confifting in the ufe of a fuperfluity of words. 



Words perfeftly fvnonimous are redundant, and ought to 

 be retrenched. Redundancy neceffarily makes the ftyle 

 weak and languid. See PleonA-sM. 



REDUNDANT Hyperbola, in Geometry, is a curve 

 of the higher kind, thus called, becaufe it exceeds the conic 

 feiSion of that name in the number of its hyperbolic legs ; 

 being a triple hyperbola, with fix hyperbolical legs. 



Redundant Interval, in Mufic, is ufed for an interval 

 exceeding the truth by a comma. 



Some apply redundant to an interval exceeding a diatonic 

 interval by a femitone minor ; but this is more ufually called 

 a fnperfluous interval. See Interval and Second. 



REDUPLICATION, in Rhetoric, a figure by which 

 a verfe begins with the fame word as the preceding one ends 

 with. See Anadiplosis. 



Reduplication, in Logic, is a kind of condition ex- 

 prefl'ed in a propofition, indicating or affigning the manner 

 in which the predicate is attributed to the fubjett. 



The ufual reduphcative words are quatenus, as, fo far as, 

 confidered as, inafmuch as, &c. Hence, 



"REDUPLICATIVE Propositions, are fuch in which 

 the fubjeft is repeated, with fome circumllances or condi- 

 tion. Thus, Men, as men, are rational ; hings, as iings, are 

 fubjeB to none but God. 



REDUTEA, in Botany, fo called by the late M. Ven- 

 tenat, in honour of his friend M. P. J. Redoute, one of the 

 moft accurate and intelligent botanical draughtfmen, and 

 perhaps the fineft botanical painter, ever known. He is 

 the author of a fplendid coloured work in folio, on the Li- 

 liaceous 



