R E C 



R E C 



! 'hat is, as much longer as tin" fide A, iu the firll reftangle, 

 i , than the fide B of the fecond reitanglc ; fo irtuch deeper is 

 the fide C in the fecond redtangle, than the fide D in the firll ; 

 and, confequently, the length of one is compenfated by the 

 depth of the other. 



Alfo, as the fide A is ^ longer than the fide C, fo the fide 

 B is I: longer than D : wherefore the reftangles muft be 

 equal. 



This is the foundatioiv of that Catholic theorem ; that the 

 redlangle of the extremes mult always be equal to that of the 

 means ; and; confequently, the rcalon of the rule of three, 

 or golden rule. 



For, fuppofe there were given any three numbers, or quan- 

 tities, geometrically proportional, as A, B, and C ; and, 

 that it were required to find a fourth, D, proportional to 

 them : fince A : B :: C : D, therefore A D = B C, and, 



T> /^ 



confequently, D = — x — j that is, the fourth term is 

 A 



equal to the quotient of the fecond, multiplied by the third 

 term, divided by the firlt. 



Or, thus in numbers : fuppofe given 12, 4, and 9, re- 

 quired a fourth proportional. Now as 12 : 4 :: 9 : Q. 



But 12 Q = 4 X 9 = 36. Therefore Q = AiLi. _ ^^ 



by dividing both fides by 1 2 . 



And hence it follows, that if any two triangles, parallel- 

 ograms, prifms, parallelepipeds, pyramids, cones, or cylin- 

 ders, have their bafes and altitudes reciprocally proportional, 

 thofe two figures or folids are equal to one another ; and, 

 •vice I'erfii, if they are equal, iheir bafcs and altitudes are re- 

 ciprocally proportional. 



Reciprocal Proportion, is when in four numbers the 

 fourth is lefs than the fecond by fo much as the third is 

 greater than the firft ; and -vice verfa. See Proportion. 



TV is is the foundation of the inverfe, or indirefil rule of 

 three. Thus, 4 : 10 :: 8 : 6. 



Great ufe is made of this reciprocal proportion, by fir 

 Ifaac Newton, and others, in demonftrating the laws of 

 motion. 



Reciprocal Theorem. See Theorem. 



RECIPROCALLY, the property of being reciprocal ; 

 thus we fay, that in bodies of the fame weight, the denfity is 

 reciprocally as the magnitude ; wz. the greater the magni- 

 tude the lefs the denfity ; and the lefs the magnitude the 

 greater the denfity ; fo again, the fpace being the lame, the 

 velocity is reciprocally as the time, and the contrary. 



RECIPROCITY. The law of reciprocity is a term 

 ufed by Legendre, in his " Theorie des Nombres," to denote 

 a reciprocal law, which has place between prime numbers of 

 different forms ; which is this ; that 711 and n being prime odd 

 numbers, 



71 — 1 

 the remainder of m - -r- « =: 



the remainder of n ' ■— m 



provided m and n are not both of the form 4 .r — ij and if 

 they are both of tliis form, then 



" — t 

 the remainder of n; ' -^ n = — 



the remainder of n ' — m, 



or they will have only contrary figns. 



RECISSION, mLaiu. See Ademption. 



REGIT, Fr, a genencal term in mufic, for what is fung 

 by a fingle voice. It is likewife applicable to inftruments ; 

 as, reck de bajfe, recti de hauthols, a folo part for the violoncello 



or hautbois. Indeed reclt, in French, feems fynonimous 

 witli the word folo in Italian, to whatever vocal or inftru- 

 mental part it is applied, in oppofition to tuiti, or chorus, in 

 which the whole band is employed. 



Redt in France is not only a technical term in mufic, but 

 in the drama, where, at the opening of a tragedy, or fub- 

 fequent to an event, it implies an account, a narration, the 

 recital of an event. In the French and Italian tragedies, in 

 imitation of the Greeks, battles and murders aro always re- 

 cited, but never tranfafted on the ilage. 



RECITAL, in Law, the rcliearful, or making mention, 

 in a deed or writing, of fomething which has been done 

 before. 



A recital is not conclufive, bccaufe it is no direft affirma- 

 tion ; and by feigned recitals in a true deed, men might make 

 what titles they pleafed, fince falfe recitals are not punifhable. 

 If a perfon, by deed of afiignment, recite that he is pof- 

 fefi'ed of an intereft in certain lands, and aflign it over by the 

 deed, and become bound by bond to perform all the agree- 

 ments in the deed ; if he is not pofleflcd of fuch interell, the 

 condition is broken ; and though a recital of itfelf is nothing, 

 yet being joined and confidered with the rcll of the deed, it 

 is material. And where it is but a recital, that before the 

 indenture the parties were agreed to do fuch a thing, it is a 

 covenant, and the deed itfelf confirms it. The recital of 

 one leafe in another, is not a fufficient proof that there was 

 fuch a leafe as is recited. But the recital of a leaje in a deed 

 of releafe, is good evidence of a leafe againft the relelfor, and 

 thofe who claim under him. A new reverfionary leafe (hall 

 commence from the delivery, where an old leafe is recited, 

 and there is none, &c. A. recites that he hath nothing in 

 fuch lands, and in truth he has an eftate there, and makes a 

 leafe to B. for years ; the recital is void, and the leafe good. 

 In this caf.', if the recital were true, the leafe would not 

 bind. 



RECITATION, the aft of reciting, or delivering a 

 difcourfe, cither in the way of narration, rehearfal, declama- 

 tion, or the like. 



RECITATIVO, Ital., RecltaUve. The Crufca Dic- 

 tionary gives no more early authority for the ule of this 

 word, as a mufical term, than that of Batifta Doni, de 

 Prreft. Muf. Veteris, publilhed in 1647 ; who defines it, 

 " a mufical compofition in an andante or plain ftyle, dif- 

 ferent from air: 'it is ufed in narrative poetry, in imitation 

 of reciting on the Ilage." 



Roufleau's definition is more full and clear. He terms it 

 " a difcourfe, or fpccch, in mufical and harmonical tones. 

 It is a melody nearly approaching to common fpeech ; a 

 mufical declamation, in which the mufician ought to imitate, 

 as much as pofiible, the inflv;xions of voice in declaiming. 

 This melody is called recitative, becaufe it refembles a nar- 

 ration, a recital ; and is ufed in the dialogue of mufical 

 dramas." 



We have prefented our readers, under the article Opera, 

 with extrafts from the prefaces of the poets and compofers 

 by whom recitative was invented, as well as from contem- 

 porary writers, who thought its origin of fufficient im- 

 portance to be recorded. 



Giovanni Batifta Doni, about the middle of the 17th 

 century, (Op. Omn. tom. ii. in Firenze, 1763, folio,) 3 

 learned and elegai.t writer on mufic, though extremely 

 warped in his judgment by a predihClion for the mufic of 

 the ancients, in a difl'crtatlon on tiie Origin of Stage-finging, 

 during his own time, gives fo curious and inftrudive an ac- 

 count of the firft operas that were performed at Florence, 

 that we ihall tranflate a part of it. 



" Some kind of cantlkna, or melody, has been introduced 



in 



