Page:Cyclopaedia, Chambers - Volume 2.djvu/587

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When two Parties have made their mutual Complaint at the fame time ; the Bulinels is firft, to determine who mall be the Accufer, and who the Accufed; i. e. on whom mail fall theisV- crimination.

Recrimination is of no force till the Criminal have been purged legally.

RECRUDESENCE, in Medicine, is when a Difeafe that was gone off returns again. See Relapse.

RECTANGLE, in Geometry, call'd alfo Oblong, and long Square, a Quadrilateral rectangular Figure, (MLIK, Tab. Geome- try, Fig. &o.) whole oppolke Sides (OP and NQ, as alfo ON and PQ_ ) are equal. See Quadrilateral.

Or, a Re&angle is a Parallelogram, whofe Sides are unequal, but Angles right. See Parallelogram.

To find the Area of a Rett angle; Meafure the length of the Sides ML and MI; and multiply them by one another: The Product is the Area of the Rectangle.

Thus ML being 545 Foot, and MI = 123 Foot; the Area will be found 42435 Square Feet.

Hence, 1°, Rectangles are in a Ratio compounded of that of their Sides ML and IK ; and therefore Rcitangles which have the fame Height, are to each other as their Bafes ; and thofe which have the fame Bafe are to each other as their Heights.

2, If therefore there be three Lines in continual Pro- portion, the Square of the middle one is equal to the Reftangle ot the two Extremes. See Proportion.

3, If there be four right Lines in continual Proportion ; the Rectangle under the Extremes is equal to the Rectangle under c'k middle Terms.

4, If from the fame Point A Fig. 41. be drawn two Lines ; one whereof, AD, is a Tangent to a Circle, the other a Secant AB : the Square of the Tangent AD, will be equal to the Rectangle under the Secant AB ; and that Part of it without the Circle, AC.

5 9, If two or more Secants ha, AB, drc. be drawn from the fame Point A; the Rectangles under their Wholes and their Parts without the Circle, will be equal. See Secant.

6", If two Chords interfect each other, the Rettangles un- der their Segments will be equal. See Chord.

RECTANGLE, in Arkhmetick, is the fame with Product or Factum. See Product and Multiplication.

RECTANGLED, Right-Angled, Triangle, is a Tri- angle, one of whofe Angles is right, or equal to 90^.

There can be but one right Angle in a plain Triangle; there- fore a reliangled Triangle, cannot be equilateral. See Tri- angle.

RECTANGLAR, in Geometry, is applied to Figures, and Solids which have one or more Angles, Right. See Angle, ffre.

Such are Squares, Rectangles, and rectangled Triangles among plain Figures; Cubes, Parallelipipeds, <&c. among Solids. See Figure, Solid, <&c.

Solids are alfo faid to be Rectangular with refpect to their Si- tuation : Thus, if a Cone, Cylinder, &c. be Perpendicular to the Plane of the Horizon, 'tis called a Rectangular or Right Cone; a Rectangular, &c. Cylinder. See Cone and Cylin- der.

The Antients ufed the Phrafe Rectangular Section of a Cone, to denote a Parabola ; that Conic Section, before Apoflonius, being only confidered in a Cone whofe Section by the Axis would be a Triangle, Right-angled at the Vertex.

Hence it was that Archimedes entitled his Book of the Qua- drature of the Parabola, by the Name of Rectanguli Coni Sectio.

RECTIFICATION, the Aft of Rectifying, i. e. of correct- ing, remedying, or redrefling fome Defect or Error, in refpect either of Nature, Art, or Morality. See Right, Rectitude, &c.

The Word is compound of rectus, right, direct, and fio, I become.

Rectification, in Chymiitry, is the repeating of a Diltil- lation or Sublimation feveral times; in order to render the Sub- ftance purer, finer, and freer from Aqueous, or Earthy Parts. See Distillation.

Rectification is a reiterated Depuration of a diltilt'd Matter, e. gr. Brandy, Spirits, or Oils ; by palling them again over their Faeces, or Marc, to render them more fublilc, and exalt their Virtues. See Spirit, &c.

Fix'd Salts are redified by Calcination, Diffolution, or Phil- tration. See Salt, Dissolution, pjrc.

Metals are rectified, i. e. refined, by the Couppel; Regulus's, by repeated Fuliorjs, &c. See Metal, Refining, &c.

Rectification, in Geometry, is the rinding of a right Line equal to a Curve. See Curve.

All we need to find the Quadrature of the Circle, is the Recti- fication of its Circumference ; it being demonftrated, that the Area of a Circle is equal to a Rectangled Triangle, whofe two Sides comprehending the right Angle, would be the Radius, and the right Line equal to the Circumference. See Circle and Circumference.

To reiiify the Circle, therefore, is to Sauare it : Or rather, both the one and the other are impoifible.

For the vaiious Attempts to rectify the Circle, in order to the Quadrature, &c. See Quadrature of the Circle.

The Rectification cf Curves is a Branch of the higher Geome- try ; wherein the ufe of the new-invented Integral Calculus, or Inverfe Method of Fluxions, is very confpicuous,— For, fince a Curve Line may be conceived to confift of innumerable right Lines, infinitely fmall ; if the Quantity of one of them be found, by the differential Calculus ; their Sum, found by the Integral Calculus, gives the length of the Curve.

Thus, fince MR=dx, mW—dy; and therefore Mot, or the Element of the Curve will be i/dx' ■}•«$».

If then, from the differential Equation, we fubftitute the Va- lue either of dx', or of dy'*, to the particular Curve, we mall have the particular Element, which being integrated, gives tha length of the Cutve. See Calculus, lntegralis and Fluxi- ons.

Indeed, the Element of the Curve is fometimes more com- modioully determined from fome particular Circumftances ; In- stances whereof we fhall give in the Rectification of the Parabola and Cycloid.

TV rectify the Parabola. For, the Parabola, we have adx—iydy

a 1 ax —ty*ay 3 "dx'—H^dy' : a

To render this Element of the Curve Integi able ; let it be re-
 * / ( Jx ' J T J >') =\/( a J'+V*' i J':a>) = a)i/(aa± 4 jy-a

folv'Vl intn an Infinity* S/nnM? f^fr Sri, tpc 1 TU- :—.

'd into an Infinite Series; .(See Series.) Theorem,

»=2 m^il V^cza 1 QrrA^ 1 : a 3 *Pm ; t.

- AQ=:a. 4;* : a*—— 2/ : a=B

— BQ=— i. ^! 4^=-i^ = c a a 1 a*

.: _ ?£. £_' _ Vl— D a* a' <? s

nn— < *>' 4/'_ ioj 1.

L>qp- T — - — ; = '—, &

a' a 1 a 7

Wherefore, dji/ (aa-^yy) ; a—dy-f-

V'ty _ lofdy

Then in a general -A

in Infinitum. zy'dy zy'dy

Whofe Integral **-. 2! _ Jl!

+

+;

 &c. Whofe Infinite expreffes the Parabolic

Arch AM. Tab. Analyfit, Fig. iS.

Hence, Firfi, Let AC, and DC (Tab. Analyfs, Fig in ) be the Conjugate Axes of an Equilateral Hyperbola; then will AC = DC = a. Suppofe MP s 2 y, QM s *; then will AP = x —a ; conlequently, by reafon PB. APsPM' xx~aa~±yy, and hence xx — $yy+aa ; confequently, x~</ Uyy-i-aa) If then jot be fuppoled Infinitely near QM, we fhall have Qa~dy and therefore the Element of the Area CQMA 5 dy ^ (aa-flyy ) The ReBification of the Parabola therefore depends on the Qua- drature of the Hyperbolic Space. '

It is to be here noted that all Integrations or Summations, are reduced to the Quadratures of Curves ; in what Cafes foeverthey be ufed, fo that to have them perfect, the Rule laid down under Quadrature cf the Logiftic Curve, muft be obferved throueh- out. 6

To Rectify the Cycloid.

Let AQ=* AB=;i, then will Q f -MS = <&, VCy-^f (x-xx). Ana hence iAP^V* 3*1 ■ 2; confequently by rea- fon of the Similitude of the Triangles AP6 and MotS, AQ,: AP : : MS : Mm x : xi : 2 : : dx : x — 1 : 2dx. Therefore Mm is the differential of the Cycloidical Arch AM

V *?7, '; 2dx - Wh «efore./5r— 1 : zdx - 2* ' : 2 - 2 AP is the Arch AM.

The Rectification of Curves Mr. tie Motive fliews may be ob- tain'd by considering the Fluxion of the Curve as an Hypothe- nufe of a Rectangular Triangle, whofe Sides are the Fluxions of the Ordinate and Abfcifl'a: Care being taken in the Expreffion of thisHypothenufe, that only oneof the Fluxions be remaining, as alfo only one of the indeterminate Quantities, viz. that whofe Fluxion is retained : an Example will render this clear.

The Right Sine CB (Fig. 20.) being given, to find the Arch Ac —Let AB = *. CB-j-. OA=r. CE the Fluxion of the Abl fciffe, ED the Fluxion of the Ordinate, CD the Fluxion of the Arch CA. From the Property of the Circle 2r*— **—,,, whence 2r.; — 2 *;- =2j>, and therefore «■ =yj. But CBj

= ; } - + ;^-„-+ r -=zr x yli-^Ji + 2UsL == _r^

rr—2rx-j-xx T rr—yy rr—yy; therefore CD =

. -^r'j—ry 4-rr— yy -

yrr~yy y'rr—yy J J_, ^

And confequently, if rr— yy be thrown into an infinite Series,

and the feveral Members of it be multiplied into rj, and then the

flowing