C Ad66)


When two Parties have made their tnutua  Conplaint at the
fame time; the Bufinefs is firfi, to determine who ihall be the
Accuier, and who the Accufed; 4. e. on whom ihall fall the Re-
criminatiox.
Reciminxatiox 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 retiangular Figure, (MLIK, Tab. Geome-
try, Fig. 6o.) whofe oppofite Sides (OP and NQ, as alfo ON
and PQ) are equal. See QUADRILATERAL.
Or, a Redangle is a Parallelogram, whofe Sides are unequal.
but Angles right. See PARALLELOGRAM.
Tofindthe Area of a Recangle; Meafure the length of the
Sides ML and Ml?; and multiply them by one another: The
Produ&c is the Area of the Rcelangle.
Thus ML being 345 Foot, and MIX 123 Foot; the Area
will be found 42435 Square Feet.
Hence, i1, Re7anwgles are in a Ratio compounded of that
of their Sides ML and IK; and therefore Redangles 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.
20, If therefore there be three Lines in continual Pro-
portion, the Square of the middle one is equal to the Redangle of
the two Extremes. See PROPORTION.
3', If there be four right Lines in continual Proportion;
the Redangle under the Extremes is equal to the .Reidaange under
the middle Terms.
4f, If from the fame Point A Fig. 4T. 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 Reaangle under the Secant AB; and that Part of it without
the Circle, AC.
,59, If two or more Secants Aa, AB, &c. be drawn from
the fame Point A; the Reaangles under their Wholes and their
Parts without the Circle, will be equal. See SECANT.
60, If two Chords interfeaft each other, the Recdangles un-
der their Segments will be equal. See CHORD.
RECTANGLE, in Arithmetick, is the fame with Produd
or Fadurn. See PRODUCT and MULTIPLICATION.
RECTANGLED, RIGHT-ANGLED, Triangle, is a Tri-
angle, one of whofe Angles is right, or equal to go9.
There can be but one right Angle in a plain Triangle; there-
fore a redangkd Triangle, cannot be equilateral.  See TRI-
ANGLE.
RECTANGLAR, in Geometry, is applied to Figures, and
Solids which have one or more Angles, 1ight. See ANGLE,
6c.
Such are Squares, Re6tangles, and rea-angled Triangles among
plain Figures; Cubes, Parallelipipeds, &c. among Solids.  See
FIGURE, SOLID, &c.
Solids are alfo faid to be Reflangular with refipeat to their Si-
tuation: Thus, if a Cone, Cylinder, &c. be Perpendicular to
the Plane of the Horizon, 'tis called a Redawgular or Right
Cone; a Reaidn&#lar, &c. Cylinder.  See CONE and CYijN-
DER.
The Antients ufed the Phrafe Reaangpr Seclion of a Cone, to
denote a Parabo  that Conic Seoion, before Apoionius, being
only confidered in a Cone whofe Se&ion 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 Redanguli Coni
Sefio.
RECTIFICATION, the Ad of Reayyfing, i. e. of corre&-
ing, remedying, or redrefling fome Defe6t or Error, in refpe&f
either of Nature, Art, or Morality. See RIGHT,RECTITUDE, &c.
The Word is compound of recdus, right, dire&s, and fib, I
become.
RECTIFICATION, in Chymiftry, is the! repeating of a Diflil-
lation or Sublimation feveral times; in order to render the Sub-
ftance purer, finer, and freer from Aqueous, or Earthy Parts.
See DISTILLATION.
Reaicatiox is a reiterated Depuration of a difiill'd Matter, e. gr.
Brandy, Spirits, or Oils; by paffing them again over their Fxces,
or Marc, to render them more fubtile, and exalt their Virtues.
See SPIRIT, &-C.
Fix'd Salts are retified by Calcination, Diffolution, or Phil-
tration. See SALT, DISSOLUTION, &C.
Metals are re~fied, i. e. refined, by the Coupel; Regulus's, by
repeated Fufions, &c. See METAL, REFINING, &c.
RECTIFICATION, in Geometry, is the finding of a right Line
equal to a Curve. See CURVE.
All we need to find the Quadrature of the Circle, is the Retii-
ficaton of its Circumference; it being demonrfrated, that the
Area of a Circle is equal to a Re&angled 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 redify the Circle) therefore) is to SqUare it: Or rather,
both the one and the other are impoffible.
For the various Attempts to redify the Circle, in order to the
Quadrature, &c. See QUADRATVRE Of thA Circle.


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The Red:iction of Curuts is a Branch of the higher GeomeS
try; wherein the ufe of the new-invented Integral Calculus, or
Inverfe Method of Flurions, is very conpicuous.-For, Mnce a
Curve Line may be conceived to confift of innumerable right
Lines, infinitely finall;i if the (quantity of one of them be founxd,
by the differential Calculus; their Sum, found by the Integral
Calculus, gives the length of the Curve.
Thus,   fince MR-i.,     R~dry; and therefore Mm, or
the Element of the Curve will be Edic' +d a
If then, from the differential Equation, we fubitute the Va-
lue either of dx , or of dy", to the particular Curve, we {hall
have the particular Element, which being integrated, gives tha
length of the Curve.  See CALCLUS, luegrabs and FLuxr.
ONS.
Indeed, the Element of the Curve is fometimes more com.
modioufly determined from fome particular Circumfiances; In-
flances whereof we flall give in the Redificatiox of the Parabola
and Cjylded.
7'd realy the Parabola.
For, the Parabola, we have adx=-2ydy
43dx'=4Y'aJ2
ad '--xt
4/ (dx' I y3)-     (dJ2+'j2 JI7) t4 'YV    (aa+4yY: *
To render this Element of the Curve Integrable; let it be re-
folv'd into an Infinite Series; 4See SERIEs.) Then in a general
Theorem,
s=2 m-l Pa2 Q=4)2: a' Pm: n=a               -A
In
AQZ-1-a. 4)"    c-ao        =
mr-n            2y   4yl     2y4
2n          +   a   a       as
m-2n CQ    3-_-   2y4 _-  _ 4Y-=D
3n               a   a'     a
m -i:2 Dig=-s±       4a%      7 i   i. in Infinitum.
47s       a    a~4   a7
~ydy   2dy
Wherefore, sd~/ (aaf+4yy): a_ d+      -Y -    -+     ±
4/ dy __ I2ydy                                      1,
a6  a-  - YX    c.  Whofe Iat*~ral+ -     *
a;        a                               3ali4
4Y"  - .!2   X &c.  Whofe Infinite expreffes the IaraboLk
Arch AM. Tab. Analafs, Fig. IS.
Hence, Fnfl, Let AC, and DC (Ta. Analyfis, FiVg. 19.) by
the Conjugate Axes of an Equilateral Hyperbola; then will
AC = DC     a. Suppofe MP     2zyQM    x   then will AP x
-a; confequently, by reafon PB. AP=PM' xx-a =4Y
and hence xx =    4yJ+aa; confequently, x  /3 y grai)   If
then Im be fuppofed Infinitely near QM, we fhail have Qq 4; d
and therefore the Element of the Area CQMA,- d y V (aa+4yy.)
The Redfificationx 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 foever they
be ufed, fo that to have them perfeft, the Rule laid down under
Xuadratare of, the Log&/lic Curve, muft be obferved through-
out.
To Redfifj the Cycloid.
Let AQ    -x ABza,    then will Qq,   MS   -dx, Pq-7  '
(x-xx). And hence: AP = Vx = xI : 2 ; consequently by rca-
fon of the Similitude of the Triangles APQ and MmS,
AQ: AP:: MS: Mm
x: xI 2 : : dx : x-I: 2dx.
Therefore Mm is the differential of the Cydoidical Arch AM
x22-: 2dx. Whereforefx-r: 2dx       a2x: 2 = 2AP is the
Arch AM.
The Reihficatiox of Curves Mr. de Moivre fliews may be ob-
tain'd by confidering the Fluxion of the Curve as an Hypothe-
nufe of a Rectangular Triangle, whofe Sides are the Fluxions of
the Ordinate and Abfciffa: Care being taken in the Expreflion
of this Hypothenufe, that only one of the Fluxions be remaining,
as alfo only one of the indeterminate Quantities, viz. that whofe
Fiuxion is retained: an Example will render this clear.
The Right Sine CB (Fig. 20.) being given, to find the Arch c.
-Let AB    =x. CB     y. OA  =r. CE the Fluxion of the Ab..
fciffe, ED the Fluxion of the Ordinate, -CD the Fluxion of'the
Arch CA.    From    the Property of the Circle 2rx-xx=.yy
whence arx -2Xx =2y;, and therefore      x =y; .   But CDa
= t~J0 i x  - O-x } ;_i                      =sr
rr-2rx+xx      rr-y;   rr-    A;
therefore CD           _     -   +    r;   r + rrfyy
vrr-Ay      n/rr-s -
And confequently, if rr-yy be thrown into an infinite Series,
=d the feveral Members of it be multiplied into r i I and then the
flowing


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