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run the fame both forwards and backwards. RECIPROCAL FIGURES, in geometry, those which have the antecedents and confequents of the fame ratio, in both figures. Thus, in plate CCXXIX. fig. 4. the fide A: B:: C: D; or 12:4:9: 3; that is, as much as the fide A, in the first rectangle, is longer than B, fo much deeper is the fide C, in the fecond rectangle, that the fide D in the first; and, confequently the greater length of the one is compenfated by the greater breadth or depth of the other; for as the fide A is

longer than C, fo B is longer than D, and the rectangles of courfe equal; that is, Ax D B x C, or 12 X 34 x9= 36.

This is the foundation of that capital theorem, viz. that the rectangle of the extremes is always equal to that of the means; and, confequently, the reason of the rule of three. See RULE.

Hence it follows, that if any two triangles, parallelograms, prifms, paralle lopipeds, pyramids, cones, or cylinders have their bafes and altitudes reciprocally proportional, thofe two figures or folids are equal to each other; and vice verfa, if they are equal, then their bafes and altitudes are reciprocally proportional. See TRIANGLE, PARALLELOGRAM,&c. RECIPROCAL PROPORTION, in arithmetic, is when, in four numbers, the fourth is less than the second, by fo much as the third is greater than the firft; and vice verfa. See the article PROPORTION. This is the foundation of the inverse, or indirect rule of three: thus, 4 : 10:: 85. See the article RULE.

Reciprocal proportion is of great use in determining the laws of motion. See the article MOTION.

RECITATIVO, or RECITATIVE, in mufic, a kind of finging, that differs but little from ordinary pronunciation, fuch as that in which the feveral parts of the liturgy are rehearsed in cathedrals; or that wherein the actors commonly deliver themselves on the theatre at the opera, when they are to exprefs fome action or paffion, to relate fome event, or reveal fone defign.

Notwithstanding this fort of compofition is noted in true time, the performer is at liberty to alter the bars of measure, and make fome long and others fhort, as his fubject requires: hence the thorough bass to the recitative is ufually plated below the other, to the end that he, who is to

accompany the voice, may rather observe and follow the finger, than the person that beats the time.

RECKONING, or a ship's RECKONING, in navigation, is that account, whereby at any time it may be known where the fhip is, and on what courfe or courfes the is to fteer, in order to gain her port; and that account taken from the logboard is called the dead reckoning. See LOG-BOARD, JOURNAL, &c.

But as the fhip's motion is liable to be difburbed from a variety of causes, fuch as the lee-way, variation of the compafs, currents, unfteadiness of the winds, &c. her place, according to the dead reckoning, may be justly doubted; and therefore mariners try every way to find the latitude their fhip is in, by observations of the fun or ftars. See the articles LEE. WAY, VARIATION, CURRENT, WIND, and LATITUDE.

Now, if the latitude found by observa tion, and that found by the dead reckoning, agree, it is prefumed the ship's place is well determined; but if they difagree, the account of longitude must be corrected and for the latitude, that found by obfervation is always to be depended on.

In correcting the longitude found by the dead-reckoning, confider whether the difference may not have been occafioned by a current; and, if poffible, make an estimate of it, as directed under the article CURRENT.

The business of correcting the dead-reckoning is a very precarious operation, and at beft is little more than gueffing; fince there may be unknown currents, occafioned by trade-winds, the tides following the moon, ftormy weather, &c. hence the beft mariners are not able to pronounce with certainty, whether the fhip may not be to the eastward or weftward of the point wherein the dead-reckoning places her.

However, the following methods are those usually taken to difcover her true place: 1. If the difference of latitude be much more than the departure, or the direct courfe has been within three points of the meridian, then the error is most likely in the distance run. 2. If the departure is much greater than the difference of latitude, or the direct course is within three points of the parallel, or more than five points from the meridian; the error may be ascribed to the courfe. 3. But if the Courses are, in general, near the middle

of

luntary imprisonment, from a motive either of devotion or penance.

The word is alfo applied to incontinent wives, whom their bufbands procure to be thus kept in perpetual imprisonment in fome religious house.

of the quadrant, the error may be either in the course, or in the distance, or in both. For to caufe an alteration in the difference of latitude, the firft of these cales requires a greater error in the course, than can well be fuppofed to have been committed in the fecond cafe, the diftances must be so faulty, as would scarce escape observation; and, in the third cafe, it is often doubtful, whether to attribute the error to the course or distance; and therefore it is usually corrected in both.

As for the methods of correcting the dead-reckoning by the variation chart, and by actually finding the fhip's true longitude from celestial obfervations, fee VARIATION and LONGITUDE. RECLAIMING, or RECLAMING, in our antient customs, a lord's pursuing, profecuting, and recalling his vaffal, who had gone to live in another place without his permiffion.

Reclaiming is alfo used for the demanding of a perfon, or thing, to be delivered up to the prince or state to which it properly belongs; when, by any irregular means, it is come into another's poffeffion.

RECLAIMING, in falconry, is taming a hawk, &c, and making her gentle and familiar.

A partridge is faid to reclaim, when the calls her young ones together, upon their fcattering too much from her. RECLINATION, of a plane, in dialling, the number of degrees, which a dial-plane leans backwards, from an exactly upright or vertical plane, that is, from the zenith.

The reclination is easily found by means of a ruler, and a quadrant; for having drawn an horizontal line on the plane by a level, or quadrant, and to it another line at right angles; apply a ruler, fo that one end of it may hang over, or reach beyond the plane: there will a quadrant, applied to the under edge of the ruler, fhew the degrees and minutes of the plane's reclination; accounting from that fide of the quadrant, that is contiguous to the edge of the ruler. RECLINER, or RECLINING DIAL. See DIAL and RECLINATION, RECLUSE, among the papifts, a perfon shut up in a small cell of an hermitage, or monaftery, and cut off, not only from all converfation with the world, but even with the house. This is a kind of vo

Reclufes were antiently very numerous : they took an oath, never to ftir out of their retreat; and having entered it, the bishop fet his feal upon the door; and the reclufe was to have every thing neceffary for the fupport of life, conveyed to him through a window. If he was a priest, he was allowed a fmall oratory, with a window, which looked into the church, through which he was to make his offerings at the mafs, hear the finging, and answer those who spoke to him; but this window had curtains before it, fo that he could not be feen. He was allowed a little garden, adjoining to his cell, in which he might plant a few herbs, and breathe a little fresh air. If he had difciples, their cells were contiguous to his, with only a window of communication, through which they conveyed neceffaries to him, and received his inftructions. If a reclofe fell fick, his door might be opened for perfons to come in and affift him, but he himlelf was not to ftir out.

F. Helyot gives a particular account of the ceremonies practiled in the reclufion of a woman, in that of mother de Cambray, in the year 1625. The bishop waited for her, early in the morning at the church-door; and upon her arrival and proftrating herself at the feet of that preJate, he gave her his benediction; conduc ted her to the grand altar, and there bleffed a mantle, veil, and fcapular, put them on her and gave her a new name. Having here made her vow, and the bishop having harangued the people in praife of the new reclufe, he conducted her proceffionally to her reclufion; the clergy all the way finging, Veni, fpoufa Chrifti, &c. Here the bishop, bleffing her afresh, confecrated the reclufion, and shut her up in perpetual confinement.

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that iffues to the juftices of the common pleas, for fending a record of a recogniz ance, which the recognizor fuggefts to have been acknowledged by force and hard dealing; in order that if it so appear, it may be annulled. RECOGNIZANCE, or RECOGNISANCE, in law, a bond or obligation of record, acknowledged to the king: thus called, because recognized or acknowledged in fome court of record, or before fome judge, mafter in chancery, or justice of the peace.

There are recognizances as well for debt, as of bail, for good behaviour, and for appearance to profecute felons, &c. which latt kinds, acknowledged before justices of the peace, are by them to be returned to the feffions, otherwise an information lies against them.

In recognizances for bail, &c. before a juffice, the principal is bound in double the fum of the fureties, the usual number of whom are two, and the penalty is 401. at leaft. Mere recognizances are not fealed, but enrolled; and execution, by force thereof, is of all the recognizor's goods or chattels (except draught-horfes and implements of hufbandiy) and the moiety of his land. The execution upon a recognizance, is termed an extent. See the article EXTENT.

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The party bound in a recognizance, is called recognizor; and the perfon to whom he is bound, is termed the recognizee. Recognizance is alfo ufed in our antient ftatutes, for the verdict of the twelve jurors upon an affile;, hence cailed recognitors. RECOIL, or REBOUND, the ftarting backward of a fire-arm, after an explosion, Merfennus tells us, that a cannon 12 feet in length, weighing 6400 th. gives a ball of 24 Ib. an uniform velocity of 640 feet per fecond. Putting, therefore, W= 6400, w 24, V640, and the velocity with which the cannon recoils we fhall have (because the momentums of the cannon and ball are equal) Wv= w V 24 × 64 w V; and fo v==2, W 6400 4; that is, it would recoil at the rate of 4 feet per fecond, if free to move. See GUNNERY and PROJECTILES. RECOLLECTION, a mode of thinking, by which ideas fought after by the mind, are found, and brought again to view. RECOLLECTS, a congregation of reformed franfciicans, called alfo friersminors of St. Francis, of the Atrict obfervance. See FRANCISCAN MONKS,

TO

RECONCILIARI, in our law-books, &c. A church is faid reconciliari, to be reconciled, when it is confecrated afresh, after having been polluted or prophaned, as by being in the poffeffion of pagans, heretics, &c.

RECONNOITRE, in war, to view and examine the ftate and fituation of things.

RECORD, an authentic teftimony in writing, contained in rolls of parchment, and preferved in a court of record. Records are faid to be of three kinds, viz. a record judicial, an attainder, &c. a record minifterial, upon oath, as an office or inquifition found; and a record made by conveyance and confent, as a fine, &c.

RECORD, among fowlers, is a bird's beginning to tune or fing, as it were within itself, or to perform its notes and difpofe its organs for finging. The cockthrush is diftinguished from the hen in recording, the firft being more loud and frequent in it than the fecond. RECORDARE FACIAS, a writ directed to the sheriff, to remove a caufe out of an inferior court, into the king's bench or common pleas. RECORDER, a perfon whom the mayor and other magiftrates of a city or corporation affociate to them, for their better direction in matters of justice, and proceedings in law; on which account this perfon is generally a counfellor, or other perfon well skilled in the law.

The recorder of London is chofen by the lord mayor and aldermen; and, as he is held to be the mouth of the city, he delivers the judgment of the courts therein and records and certifies the city cuftoms.

RECORDO ET PROCESSO MITTENDIS, is a writ to call a record, together with

the whole proceedings in a caufe, out of an inferior court into the king's court. RECOVERY, in law, is obtaining any thing by judgment or trial at law.

Recoveries are of two kinds, a true recovery and a feigned or common one. A true recovery is the actual recovering of any thing, or its value, by judgment and trial at law: as where a perlon is fued for land, or other things real and perfonal, and obtains a verdict in his favour. A feigned or common recovery, is a formal act by confent, made use of for the better fecuring of lands, tenements, &c. the end and effect of which is, to dock and deftroy estates-tail, remainders, and reversions,

reverfions, and to bar the former owners. In a common recovery, there must be at leaft three parties, viz. the demandant, tenant, and vouchee: the demandant is the person that brings the writ of entry, and therefore may be termed the recoveror; the tenant is he against whom the writ is brought, who may be termed the recoveree; and the vouchee is the perfon whom the tenant vouches, or calls to warrant for the lands demanded: thus, when a person is defirous to cut off an eftatetail in lands, &c. he caufes a feigned writ of entry sur disseifin en le port to be brought by fome friend, who is the demandant, for thofe lands, &c. who in a feigned declaration thereupon made, pretends that he was diffeifed by him, who, by a feigned fine, or deed of bargain and fale, is named and fuppofed to be the tenant of the land: this feigned tenant, if it be a fingle recovery, is made to appear and vouch to warranty the crier of the court of common pleas, or the bagbearer of writs to the cuftos brevium in that court, who is termed the common vouchee, and is supposed to warrant the title; but he making default, a judgment is by this fiction entered, that the demandant hall recover, and have a writ of feifin for the poffeffion of the lands in question; and that the tenants shall recover the value of the lands against the common vouchee: though this recovery in value is only imaginary, yet it is looked upon as a bar to the intail for

ever.

RECREMENT, in chemistry, fome fuperfluous matter feparated from fome other that is useful in which fenfe it is the fame with fcoriæ, fæces, and excrements, See the article SCORIA, &c. RECRIMINATION, in law, an accufa tion brought by the accufed against the` accufer, upon the fame fact. See the article ACCUSATION.

RECRUITS, in military affairs, newrailed foldiers, defigned to fupply the place of thofe who have loft their lives in the fervice, or are difabled by age or wounds. See the article SOLDIER. RECTANGLE, in geometry, the fame with a right angled parallelogram. See the article PARALLELOGRAM.

In arithmetic and algebra, a rectangle gnifies the fame with fa&tum or product. See the articles PRODUCT and MULTIPLICATION.

RECTANGLED, RECTANGULAR, or RIGHT-ANGLED, appellations given to

figures and folids which have one or more right-angles: thus a triangle with one right angle, is termed a rectangled triangle; also parallelograms with right angles, fquares, cubes, &c. are rectangular.

Solids, as cones, cylinders, &c. are alfo faid to be rectangular, with respect to their situation, when their axis are perpendicular to the plane of the horizon. The antient geometricians always called the parabola, the rectangular fection of a cone. See the articles CONIC SECTIONS and PARABOLA. RECTIFICATION, the art of fetting any thing to rights: and hence, to rectify the globes, is to fit them for performe ing any problem. See GLOBE. RECTIFICATION, in geometry, is the finding a right line, equal in length to a curve. See the article CURVE,

The rectification of curves is a branch of the higher geometry, where the use of the inverfe method of fluxions, is very confpicuous.

Cafe I. Let A C G (pl. CCXXIX. fg. 3. n° 1.) be any kind of curve, whose ordinates are parallel to themfelves, and perpendicular to the axis AQ. Then if the fluxion of the abfcifs AM be denoted by Mm, or by Cn, (equal and parallel to Mm) and nS, equal and parallel to Cr, be the reprefentation of the corresponding fluxion of the ordinate MC; then will the diagonal CS, touching the curve in C, be the line which the generating point p, would defcribe, were its motion to become uniform at C; which line, therefore, truly expreffes the fluxion of the fpace AC, gone over.

See the article FLUXION.

Hence, putting AM = x, CM = },
and AC = x; we have ≈ (=CS=
√ Cn2+Sn2=√x2+2; from which,
and the equation of the curve, the value
of z may be determined. Thus, let
the curve proposed be a parabola of
any kind, the general equation for
which is x
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Cafe II. Let all the ordinates of the propofed curve ARM (ibid. n° 2.) be referred to a center C: then, putting the tangent RP (intercepted by the perpendicular CP)t, the arch, BN, of a circle, defcribed about the center C,=x; and the radius CN (or C B) = a; we havey::y (CR): (RP); and, confequently, from whence the vaż =

lue of z may be found, if the relation of y and t is given. But, in other cafes, it will be better to work from the followた

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a2

ing equation, viz. ż= which is thus derived; let the right line CR, be conceived to revolve about the center C; then fince the celerity of the generating point R, in a direction perpendicular to CR, is to (x) the celerity of the point N, as CR () to CN (a), it will therefore be truly represented by; which being to (ỷ) the ce

a

lerity in the direction of CR, produced as CB (s): RP (t), it follows that

a2 22:12, whence, by compofition, $ 2x2 + j2 : j2 :: s 2 + t2 (y2) : ƒ 2; there

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But the fame conclufion may be more eafily deduced from the increments of the flowing quantities: for, if R m, rm, and Nu be affumed to reprefent (x, y, x) any very small correfponding increments of AR, CR, and BN; then will CN (a): CR(y): (the arch N): the yx a

fimilar arch Rr. And if the tri

angle R r m (which, while the point m is returning back to R, approaches continually nearer and nearer to a fimilitude with CRP) be confidered as rectilinear, we shall alfo obtain 2 (=

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2, as before.

Now from the right fine, verfed fine, tangent, or fecant of an arch of a circle OARB (ibid. n° 3.) given: to find the length of the arch itself, in terms thereof. Put the verfed fine AB=x, the right fine Rby, the tangent ATt, the fecant OTs, the arch AR≈, and the radius AO, or RO = a; also let Rnx, nr=y, and Rr: then, fince LrnR (a right angle)=LOьR, and rRn OR b, the triangles r Rx, and ORb, are equiangular; and Rb(j): OR (a) :: Rn() : Rr (ż)=

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