R E C 



fimple, in form of a fquarc, or rather a bevel (which lee) ; 

 confiding of two arms, or branches, rivetted together, and 

 yet moveable, like a fcdor, on the centre or rivet. 



To take an angle with it, they lay the centre of a pro- 

 traftor to the joint, and the degrees cut by the edge (hew 

 the quantity of the angle : othcrwife the angle made by the 

 twQ rulers is drawn on paper, and then meafured with a pro- 

 traftor. 



Sometimes there is a circle divided into degrees added over 

 the centre or rivet, with an index to (liew the degrees with- 

 out a protraftor. At other times the under branch is 

 divided. 



To meafure a faliant angle with any of the rccipiangles, 

 apply the infides to the lines that form the angle ; for a re- 

 entering angle, apply the outfides, &c. 



RECIPIENDO Excommunicato. See Excommuni- 

 cato. 



Recipiendo et fac'iendo attoinnto. See Attoknato. 

 RECIPIENT, in Chemljry. See Receiver. 

 Recipient, Ilnnan, a vefiel like a tea-pot, intended for 

 the feparation of elfential oils from the watery liquor on 

 which they float. 



Recipient of an Jir-pump. See Receiver. 

 RECIPROCAL, Reciprocus, fomething that is mu- 

 tual, or which is returned equally on both fides, or affeds 

 both parties alike. 



The end of human fociety is to afford each other recipro- 

 cal aid ; there are reciprocal duties between the prince and 

 his fubjecfts, the hufband and wife, &c. There is a recipro- 

 cal aftion between the agent and patient. 



The lex talionis eftablifties a kind of reciprocation of 

 juftice. 



If two fimilar triangles be cut by parallel lines, the 

 feftions of the fides will be proportional ; and recipro- 

 cally, if the fides be cut proportionably, the triangles are 

 fimilar. 



Reciprocal, in Lo/tc, is applied to terms which have 

 the fame fignification, or are convertible ; as, reafonabk animal 

 and man. 



Schoolmen define reciprocation a converfion of the fe- 

 veral terms in an enunciation. And terms are faid to be 

 converted in an enunciation, when the predicate is put in 

 the place of the fubjeft, and, reciprocally, the fubjeft in that 

 of the predicate. 



Thus, rationality andrifibility are faid to reciprocate ; for 

 we fay equally, a rational is rijible, and a rijible is ra- 

 tional. 



Reciprocal, in Grammar, is applied to certain verbs 

 and pronouns in fome of the modern languages ; in regard 

 of their turning or reflecting the noun, or perfon, upon 

 himfelf. 



Thus the pronoun relative, himfelf, refers Cato to Cato's 

 felf. 



The abbe de Dangeau defines reciprocal verbs to be thofe 

 whofe nominative is plural, and denotes perfons afting mu- 

 tually on one another : as, Ces quatre hommes s^entrebattoient, 

 thefefour men fought together; Pierre et toi votis vous louez, 

 Peter and you praife one another, &c. 



Reciprocal verbs are a fpecies of thofe which that author 

 calls pronominals, and which he diflinguiihes into reciprocal 

 and identical. 



Reciprocal, in Poetry, is applied to verfes which run 

 the fame both backwards and forwards ; called alfo recurrents. 

 See Retrograde. 



Reciprocal, in Arithmetic and Algebra, is the quotient 

 arifing fr«m the divifion of unity by any number or quan- 



R E C 



t'ty. Thus, the reciprocal of a = — , the reciprocal of — = 



a J 



y 



-, and fo on. 



Reciprocal Equations, in Algebra, are thofe equations 

 which contain feveral pairs of roots which are the recipro- 

 cals of each other. Thus, an equation wliofe roots are a, — ; 



b, ~; &c. is called a reciprocal equation. 



Some authors 



define reciprocal equations, to be thofe equations whofe co- 

 eificients proceed in the fame order from both extremes, and 

 refpeftively equal to each other : thus, x" ■\- ax^ -f- i x' 4- 

 ix' -|- a.v + I = o, is a reciprocal equation ; but this form 

 ought rather to be confidered as a neceffary property of thefe 

 equations, by which they may be readily diilinguHhed, than 

 to be employed in the definition of them ; it being doubtlefs 

 the reciprocity of the roots from which they have received 

 their peculiar appellation. 



The folution of thefe equations may always be made to 

 depend ujjon others of half the original degree, when the 

 equation is of even dimenfions ; or upon half the dimenfion 

 minus I , when it is odd. 



Thus far, in faft, this property is not exclufively due to 

 reciprocal equations, as the fame may be done in all cafes 

 where any fimilar relation is known to have place between the 

 roots of an equation, whether by multiplication, divifion, ad- i 

 dition, or fubtraftion ; but in thefe cafes there is nothing in 

 the form of the equation by which fuch relations may be 

 known to have place, whereas in reciprocal equations their 

 reciprocity becomes immediately obvious by the peculiar 

 order of their co-efficient, thefe being the fame from both ex- 

 tremes, both with regard to fign and magnitude. Hence 

 any equation of the form 



x" +<»«"■"' + bx"-'^ +....^x^-|-<3x+ 1 = 



is immediately known to be reciprocal, and we may proceed 

 in its folution as follows. 



Firft, if the equation be of odd dimenfions, as 



«'" + '+ a x"-" + bx'"-' + ....bx^+ax-j-l = o, 

 then it is obvious that either -f i, or — i, is one of its 

 roots, for either -|- 1, or — i, fubftituted for x, according 

 as the figns of the co-efficients may require, will obvioufly 

 render the whole expreffion equal to zero, and is therefore 

 a root of the equation. The equation may, therefore, be 

 immediately reduced to another of lower dimenfions by di- 

 vifion, according to the known theory of equations ; thus, 

 let the propofed equation be 



k' — 5 *' + 7 •■«' + 7 ■•«' — 5 * + 1=0- 

 Here x = — i is obvioufly one of the roots, as this fubfti- 

 tuted for X renders the whole equal to zero ; and confe- 

 quently the whole equation is divifible by .v -f- I, according 

 to the known theory of equation, which divifion being 

 made, we have, for our reduced equation, 



x^ — 6 -v^ + 1 3 x' — 6 X -)- 1 = 0, 



which is now of even dimenfion one degree lefs than the 

 original equation, and fl:iU reciprocal. We need, therefore, 

 only confider equations of the latter form ; -uiz. thofe of 

 which the index of the higheft power is an even number. 



Let x"+ /x-'-'-l-jx"-" + &c. jx'-f-/x-(- I = o 

 be any reciprocal equation, whofe roots are a, — ; ^j -r » 



