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tt {he Lines be equidiflant, and it cut 'em  at right Angles,   Again, as
a ltypethola of the firfi Kind has two
it is call'd the Axis 5 and the Point A, whence the piame- totes; that of
the fecond has three, that of the thi
ter is drawn, is call'd the Vertex. See DIAMETER, Axis, Eec. and as the Parts
of any right Line between th
and VERTE'X.                                              Hyperbola and its
two Afyiiiptdtes are equal on eitr
The equidiflant Lines M M are call'd Ordinates, or Ap- io, In Hyperbolas
of the fecond Kind, any right L
i5licates  and their Halves, PM, Semiordinates. See OR- ting the Carve and
its three Afymrptotes in three


DINATE} and SEMIORDINATYE-                            the Sum of the twoParts
of that right Line, exten.ed
The Portion of the Diameter AP, between the Vertex, any two Afymptotes the
fame tvay to two Points of the
or other lix'd Point. and an Ordinate, is call'd the Abfcifce. Curve, is
equal to the third Part, extended from the third


See ABscissE.                                      Afymptote, the contrar
way to the third Pointof theCa
AmA the C      f of ai thef Diameters. the Centre. See See ASYNIPTOTE, HYBERBOLA,
SC.
.Lfla  tnewotuul  a p    -a ,, ..- -.,.[_-


CENTRE.                                                 Again, as in other
Conic Sscfions, ijot parabolica
Curvcs are diffinguifh'd into Algebraic, frequently with  Square of the Ordinate,
i. e. the Refaangle of the
5Des Cartes call'd Geometrical; and Tlranflcendental, called  nates, drawn
to contrary Parts of the Diameter, is X
by the fame Cartes, &c. Mechanical Curves.            Redangle of the
Parts of the Diameter terminated
Algebraical CURVEs, are thofe wherein the Relation of Vertices of an Ellipfis
or Hyperbola, as a given Line,
the Abfciflhs A P. A P, AP, to the Semiordinates M P, the Latus Redtm, is
to that Part of the Diameter
M P, M P, may be exprefs'd by an Algebraical Equation. lies between the Vertices,
and is call'd the Latus Bra
See EQUATION.                                      I llm    So, in Curves
of the fecond Kind, not parab
Suppofe, v. g. in a Circle, (Fig. 52.) A B-a, A Px, the Parallelopiped under
the three Ordinates, is to tb
PM    y; thenwill PB =a-x; confequently, as PM'=      rallelopiped under
the Parts of the Diameter cut off
A P, P By'= a x-x'. Or, fuppofe P C c x, A C- a, P M  Ordinates and the three
Vertices of the Figure, in a
_y; then will M C'-PC'- P M; that is, a'-x'z y'. Ratio: wherein, if there
be taken three right Lines
Note, Thofe are alfo call'd Algebraic Curves, which are three Parts of the
Diameter plac'd between the Verti
of a determinate Order; fo, as that the Equation always the Figure, each
to each; then thofe three right Line
continues the fame in the feveral Points of the Curve.  be call'd the Latera
Reda of the Figure, and the P
Mofi Authors, after lDes Cartes, call Algebraic Curves, the Diameter between
the Vertices, the Latera 2lran
Geometrical ones; as admitting none elfe into the Confiruc-  And, as in a
Conic Parabola, which has only one-s
tion of Problems; nor, confequently, into Geometry. But to one and the fame
Diameter, the Refangle und
ol'fius, from Sir 1. Newton, and M. Leibnitz, is of ano- Ordinates, is equal
to the Redangle under the Part
ther Opinion; and thinks, that in the Conflruaion of a Diameter cut off at
the Ordinates and Vertex, and a
Problem, one Curve is not to be preferr'd to another, for its right Line
call'd the Latus Retgum: So, in Curves
being defin'd by a more fimple Equation, but for its being fecond Kind, which
have only two Vertices to the fame
more eafily defcrib'd. See PROBLEM.                   meter the Parallelopiped
under three Ordinates, is ec
A Y'ranfcendental CuRvE, is that which can't be defin'd the Parallelopiped
under two Parts of the Diamete
by an Algebraic Equation.                             offat the Ordinates
and the two Vertices, and a giver
Thefe Curves, Des Cartes, &c. call A1echanical ones; Line, which may
therefore be call'd the Latus Yra
and under that Notion exclude 'em out of Geometry: But finm. See LATUi; fee
alfo PARABOLA.
Newton and Leibnitz, for the Reafon abovemention'd, are  Further, as in the
Conic Se&ions, where two Paralle


of another Opinion.  Indeed, Lezlnitz has tound a new  minated on each lide
by a Uurve, are cut by two rarali
Kind of Equations, which he calls 7Iranfcendental Equa- terminated on each
fide by a Curve; the firfi by the th
tions; whereby even Y'ranfcendental Curves, and thofe and the fecond by the
fourth : the Reaangle of the P
which are not of any determinate Order, i. e. which don't of the firfl, is
to the Redangle of the Parts of the feco
continue the fame in all the Points of the Curve, may be as that of the fecond
is to that of the fourth  So, wi


defin'd. Ati. Erudit. LeiibP. A. I684. P. 134.        four fuch right Lines
occur in a Curve of the fecond Kind,
Algebraic CURVES of the fame Kind or Order, are thofe each in three Points;
the Parallelopiped of the Parts of the
whofe Equations rife to the fame Dimenfion. See ORDER.  firff, will be to
that of the Parts of the fecond, as that of
Geometrical Lines being defined by the Relation between  the fecond to the
Parts of the fourth.  See SECTION.
the Ordinates and Abfciffes, or (which is the fame, by the  Lafily, the Legs
of Curves, both of the firf, fecond, anct
Number of Points wherein they may be cut by a right Line) higher Kinds, are
either of the Parabolic or Hyperbolic
are well diflinguith'd into two Kinds or Orders: In which  Kind: an Hyperbolic
Leg, being that which approaches
view, Lines of the firil Order will be right Lines; and thofe infinitely
towards fome Afymptote; a Parabolic, that which
of the fecond or quadratic Order will be Curves, viz. Conic  has no Afymptote.
See ASYMPTOTE.
Setlions.                                                Thefe Legs are befi
diflinguifli'd by their Tangents; fbr,
Now, a Curve of the firfi Kind is the fame with a Line  if the Point of Contaa
go off to an infinite Difiance, the
of the fecond, (a right Line not being number'd among  Tangent of the Hyperbolic
Leg, will coincide with the A-
Curves) and a Curve of the fecond Kind, the fame with a fymptote ; and that
of the Parabolic Leg, recede infinitely,
Line of the third.  Thus, Curves of the firfi Kind, are and vanillh. The
Afymptote, therefore, of any Leg, is
thofe whofe Equation rife to two Dimenfions; if they rife found by feeking
the Tangent of that Leg to a Point ininite-
to three, the Curves are of the fecond Kind, if to four, of ly diflant; and
the Bearing of an infinite Leg, is found by
the third, &c.                                        feeking the Pofition
of a right Line parallel to the Tangent,
Thus, e. g. the Equation for a 'Circle is, y' =ax -X2,  when the Point of
Conta&t is infinitely remote: for this Line
or a'-x'y2. A Circle, therefore, is a Curve of the    tends the fame way
towards which the infinite Leg is din
firfi Kind.                                           reaed.
Again, a Curve of the firfit Kind, is that defined by the   Redugion of CURVES
of the fecond Kind.
Equation ax= y'; and a Curve of the fecond Kind, that    Sir I. Newtton reduces
all Curves of the fecond Kind to
defined by the Equation az x =y'. See CIRCLE.          four Cafes of Equation:
In the firfi, the Relation between
For the various Curves of the firft Kind, and their Pro- the Ordinate and
Abfciffe, making the Abfciffe x, and the
perties, fee CONIC Segions.                            Ordinate y, affumes
this form xyy+ey=ax'+bxx+
For Curves of the feeond Kind, Sir I Newton has a difg cx+d. -n the fecond
Cafe, the Equation affumes this
tin& Treatife, under the Title of Enumeratio linearum ter- form xy=ax'+bx'
+cx+d.     In the third Cafe, the
iii ordinis.                                          Equation is yy=axl~bx'+cx+d.
         In the fourth, the
Curves of t1hefecond and other higher Kinds, he obferves, Equation is of
this form, y =a x + b x2c+ cx +d.
have Parts, and Properties fimilar to thofe of the firf-:
Thus, as the Conic Seffions have Diameters and Axes; the   Enumeration of
the CURVES of the fecond Kind.
Lines cut or biffeaed by thefe, are call'd Ordinates; and the  Under thefe
four Cafes, the fame Author brings a vaA
Interfeffion of the Curve and Diameter, the Vertex: So, in  Number of different
Forms of Curves, to which he gives
Curves of the fecond Kind, any two parallel right Lines be- different Names.
ing drawn fo as to meet the Curve in three Points; a right  A Hyperbola lying
wholly in the Angle of the Afymp.
Line cutting thefe Parallels, fo, as that the Sum of the two totes, like
a Conic Hyperbola, he calls an Infcribed Hyper. -
Parts between the Secant and the Curve on one fide, is equal bola; that which
cuts the Afymptotes, and contains the
to the third Part terminated by the Curve on the other Parts cut off within
its own Periphery, a Circumfcrib'd Hy-
fide, will cut, in the fame manner, all other right Lines perbola; that,
one of whofe infinite Legs is infcrib'd, thy
?arallel to thefe, and that meet the Curve in three Points, other circumfcrib'd,
he calls Ambiginal; that whofe Legs
s. e. fo, as that the Sum of the two Parts on one fide, will look towards
each other, and are direded the fame way,
be fill equal to the third Part on the other fide.    Converging; that where
they look contrary ways, Diverg-
Thefe three Parts, therefore, thus equal, may be call'd  Cng ig that where
they are convex difFerent ways, Croig-legX?
Ordinates, or Applicates; the Secant the Diameter  and  that'applied to its
Afymptote, with a concave Vertex, and
where it cuts the Ordinates at right Angles, the Axis : The  diverging Legs,
Conchoidal; that which cuts its Afymptoto
Interfe~tion of the Diameter and the Curve, the Vertex; with contrary Flexures,
and is produced each way into con-
and the Concourfe of the two Diameters, the Centre X and  trary Legs, A.dguineous,
or Snake-like; that which cuts its
tie Concourfe of all the Diameters, the General Centre.                 
                         Conjugate


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