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direction of a fine. That is to say, that when 
a fine is levied, or a recovery is suffered, a 
deed is made between the parties really inte- 
rested, which declares the purposes of tlie 
fine or recovery, and this deed is called a 
deed to lead or to declare the uses according 
as it is made before or after the fine or re- 
covery. To enter at full into the learning 
of fines and recoveries would be impossible 
in a general dictionary. It is sufficient to 
say that both of them are in the nature of a 
sham suit, while one of which is compro- 
mised and the other carried on to judgment 
by default between the parlies really inte- 
rested, and the use of them is to enable a 
married woman to make a good convey- 
ance, and a tenant in tail to turn his estate 
into an estate in fee, or as it is called, to 
dock or bar the entail. See Fine and 
Estate. 
RECTANGLE, in geometry, the same 
with a right-angled parallelogram. In arith- 
metic and algebra, a rectangle signifies the 
same with factum or product. 
RECTANGLED, Rectangular, or 
Right-ang led, appellations given to figures 
and solids which have one or more right- 
angles : thus a triangle with one right angle, 
is termed a rectangled tringle ; also paralle- 
lograms with right angles, squares, cubes, 
&c. are rectangular. Solids, as cones, 
cylinders, &c. are also said to be rectan- 
gular, with respect to their situation, when 
their axis are perpendicular to the plane of 
the horizon. The ancient geometricians al- 
ways called the parabola, the rectangular 
section of a cone. 
RECTIFICATION, the art of setting 
any thing to rights : and hence, to rectify 
the globes, is to fit them for performing 
any problem. 
Rectification, in geometry, is the find- 
ing a right line, equal in length to a curve. 
The reotificatipn of curves is a branch of 
the higher geometry, where the use of the 
inverse method of fluxions is very conspicu- 
ous, of which we shall give an example. 
Case I. Let A C G, (Plate Miscel. XIII, 
fig. 15) be any kind of curve, whose ordinates 
are parallel to themselves, and perpendicular 
to the axis A Q. Then if the fluxion of the 
absciss A M be denoted by M m, or by C n, 
(equal and parallel to M m) and n S, equal 
and parallel to Cj', be the representation 
of the corresponding fluxion of the ordinate 
M C ; then will the diagonal C S, touching 
the curve in C, be the line which the gene- 
rating point p, would describe were its mo- 
tion to become uniform at C ; which line, 
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therefore, tnily expresses the fluxion of 
the space A C, gone over. Hence, putting 
A M = T, C M =: y, and A C = 2 ; we have 
55 (— C S = ^ C -|- S K* =2 i 
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 
i and hence x = , and there- 
fore (:=z \/ \X =: 
,'Zn— i'l I 
the fluent of which, uni- 
versally expressed in an infinite series, is y -f- 
— 3 
— 1 X 4« — 3 X 
Case II. Let all the ordinates of the pro- 
posed curve A R M (fig. 16), be referred 
to a centre C : then putting the tangent 
R P (intercepted by the perpendicular CP) 
= t, the arch, BN, of a circle, described 
about the centre C, = a; j and the radius 
CN(or C B) = a; we have z:y ::y{C'R) 
: f (R P) ; and, consequently, z : from 
whence the value of r may be found, if the 
relation of y and t is given. But, in other 
cases, it will be better to work from the 
following equation, viz. z = \/ _i_ 
-T r r 
which is thus derived; let the right line 
CR, be conceived to revolve about the 
centre C; then since the celerity of the 
generating point R, in a direction perpen- 
dicular to C R, is to (x) the celerity of the 
point N, as C B (y) to C N (a), it will there- 
fore be truly represented by ^ ; which 
being to ( j) the celerity in the direction of 
C R, produced as C B (s) : R P (f), it fol- 
lows that : P ; whence, by 
composition, -[- P (y^) : 
P ; therefore = ^P ’ conse- 
quently •^-l-y^(=^)=».Q.E.D, 
But the same conclusion may be more 
easily deduced from the increments of the 
flowing quantities : for, if R m, r m, and 
N n be assumed to represent («, j, x) any 
