CUR
C 360)
CUR
It the Lines be equidiftant, and it cut 'em at right Angles, it is call'd the Axis ; and the Point A, whence the Diame- ter is drawn, is call'd the Vertex. See Diameter, Axis, and "Vertex.
The equidiftant Lines MM arc call'd Ordinates, or Ap- flicates ; and their Halves, PM, Semiordinatcs. See Or- dinate, and Semiordinate.
The Portion of the Diameter AP, between the Vertex, or other fix'd Point, and an Ordinate, is call'd the Abfciffe. See Auscisse.
And the Concourfe of all the Diameters, the Centre. See
Centre. , ,
Curves are diftinguifh'd into Algebraic, frequently with
Des Cartes call'd Geometrical-, and Tranfcendental, called
by the fame Carres, &c Mechanical Curves.
Algebraical Curves, are thofe wherein the Relation of the AbCcifles AP, AP, AP, to the Semiordinates MP, MP, MP, maybe exprefs'd by an Algebraical Equation. See Equation.
Suppofe, v.g. In a Circle, (Fig. 52.) AB=:^, AP = .v, PM^^j then will VB=a — *; confequently, as PM J — AP, YBy' — ax—x*. Or, fuppofe PC = .v, AC=^,PM = y; then will MC 3 =PC 2 = PM* 5 that is, «"— a?*==jj?'. Note, Thofe are alio call'd Algebraic Curves, which are of a determinate Order 5 fo, as that the Equation always continues the fame in the feveral Points of the Curve.
Moft Authors, after 2)e$ Cartes, call Algebraic Curves, Geometrical ones ; as admitting none elfe into the Comtruc- tion of Problems ; nor, confequently, into Geometry. But Wolfius, from Sir I. Newton, and M. Leibnitz, is of ano- ther Opinion ; and thinks, that in the Conftruction of a Problem, one Curve is not to be preferr'd to another, for its being defin'd by a more fimple Equation, but for its being more eafily defcrib'd. See Problem.
ATranfcendentalCuRW, is that which can't be defin'd by an Algebraic Equation.
Thefe Curves, 'Des Cartes, &c. call Mechanical ones 5 and under that Notion exclude 'em out of Geometry : But Newton and Leibnitz, for the Reafon abovemention'd, are of another Opinion. Indeed, Leibnitz has found a new Kind of Equations, which he calls c tranfce?idental Equa- tions ; whereby even T'ranfcendental Curves, and thofe which are not of any determinate Order, i.e. which don't continue the fame in all the Points of the Curve, may be defin'd. A£i. Erudit. Leip. A. 11584. p- a 34-
Algebraic Curves of the fame Kind or Order, are thofe whole Equations rife to the fame Dimenfion. See Order. Geometrical Lines being defined by the Relation between the Ordinates and Abfciffes, or (which is the fame, by the Number of Points wherein they may be cut by a right Line) are well diftinguifli'd into two Kinds or Orders : In which ■view, Lines of the firfl: Order will be right Lines ; and thofe of the fecond or quadratic Order will be Curves, viz. Conic Sections*
Now, a Curve of the firfl: Kind is the fame with a Line of the fecond, (a right Line not being number'd among Curves) and a Curve of the fecond Kind, the fame with a Line of the third. Thus, Curves of the firfl: Kind, are thofe whofe Equation rife to two Dimenfions ■ if they rife to three, the Curves are of the fecond Kind 5 if to four, of the third, &c.
Thus, e.g. the Equation for a 'Circle is, y* — a x — x*, or a 3 — x 3 =y 2 - A Circle, therefore, is a Curve of the firit Kind.
Again, a Curve of the firji Kind, is that defined by the Equation ax=.y* ; and a Curve of the fecond Kind, that defined by the Equation a*x=y*. See Circle.
For the various Curves of the fir Jl Kind, and their Pro- perties, fee Conic Sections.
For Curves of the fecond Kind, Sir I. Newton has a dif- tincl Treatife, under the Title of Entime ratio linearum ter- tii ordinis.
Curves of the fecond and other higher Kinds, he obferves, have Parts, and Properties fimilar to thofe of the firit : Thus, as the Conic Sections have Diameters and Axes; the Lines cut or biffected by thefe, are call'd Ordinates ; and the Interferon of the Curve and Diameter, the Vertex : So, in Curves of the fecond Kind, any two parallel right Lines be- ing drawn fo as to meet the Curve in three Points ; a right Line cutting thefe Parallels, fo, as that the Sum of the two Parts between the Secant and the Curve on one fide, is equal to the third Part terminated by the Curve on the other iide, will cut, in the fame manner, all other right Lines parallel to thefe, and that meet the Curve in three Points, 1, e. fo, as that the Sum of the two Parts on one fide, will be ftill equal to the third Parr on the other fide.
Thefe three Parrs, therefore, thus equal, may be call'd Ordinates, or Applicates ; the Secant the Diameter ; and where it cuts the Ordinates at right Angles, the Axis : The Interferon of the Diameter and the Curve, the Vertex 5 and the Concourfe of the two Diameters, the Centre ; and the Concourfe of all the Diameters, the General Centre.
Again, as a Hyperbola of the firfl: Kind has two Afymrrf totes - y that of the fecond has three, that of the third four &c. and as the Parts of any right Line between the Conic Hyperbola and its two Afymptotes arc equal on cither fide- fo, in Hyperbolas of the fecond Kind, any right Line cut- ting the Curve and its three Afymptotes in three Points • the Sum of the two Parts of that right Line, extended frora any two Afymptotes the fame way to two Points of the Curve, is equal to the third Part, extended from the third Afymptote, the contrary way to the third Point of thtCurve. See Asymptote, Hybereola, ££?c.
Again, as in other Conic Sections, hot parabolical, the Square of the Ordinate, i.e. the Rectangle of the Ordi- nates, drawn to contrary Parts of the Diameter, is to the Rectangle of the Parts of the Diameter terminated at the Vertices of an Ellipfis or Hyperbola, as a given Line, call'd the Latus Re&um, is to that Part of the Diameter which lies between the Vertices, and is call'd the Latus Tranfber* fum : So, in Curves of the fecond Kind, not parabolical, the Parallclopiped under the three Ordinates, is to the Pa- rallelopiped under the Parts of the Diameter cut off at the Ordinates and the three Vertices of the Figure, in a given Ratio : wherein, if there be taken three right Lines at the three Parts of the Diameter plac'd between the Vertices of the Figure, each to each 5 then thofe three right Lines may be call'd the Latera Re&a of the Figure, and the Parts of the Diameter between the Vertices, the Latera Iranfverfdt And, as in a Conic Parabola, which has only one'Vertex to one and the fame Diameter, the Rectangle under the Ordinates, is equal to the Rectangle under the Part of the" Diameter cut off at the Ordinates and Vertex, and a given right Line call'd the Latus Re&um: So, in Curves of the fecond Kind, which have only two Vertices to the fame Dia- meter ; the Parallclopiped under three Ordinates, is equal ro the Parallclopiped under two Parts of the Diameter cut off at the Ordinates and the two Vertices, and a given right Line, which may therefore be call'd the Latus 'Tranfver- fum. See Latus ; fee alfo Parabola.
Further, as in the Conic Sections, where two Parallels ter- minated, on each fide by a Curve, are cut by two Parallels, terminated on each fide by a Curve ; the firit by the thirds and the fecond by the fourth : the Rectangle of the Parts of the firfl 1 , is to the Rectangle of the Parts of the fecond, as that of the fecond is to that of the fourth : So, when four fuch right Lines occur in a Curve of the fecond Kind, each in three Points; the Parallclopiped of the Parts of the firfl:, will be to that of the Parts of rhe fecond, as that of the fecond to the Parts of the fourth. See Section.
Laftlv, the Legs of Curve s, both of the firfl, fecond, and higher Kinds, are either of the Parabolic or Hyperbolic Kind : an Hyperbolic Leg, being that which approaches infinitely towards fome Afymptote; a Parabolic, that which has no Afymptote. See Asymptote.
Thefe Legs are belt diitinguifh'd by their Tangents ; for, if the Point of Contact go off to an infinite Distance, the Tangent of the Hyperbolic Leg, will coincide with the A- fymptote 5 and that of the Parabolic Leg, recede infinitely, and vanifh. The Afymptote, therefore, of any Leg, is- found by feeking the Tangent of that Leg to a Point infinite- ly diftant ; and the Bearing of an infinite Leg, is found by feeking the Pofition of a right Line parallel to the Tangent, when the Point of Contact is infinitely remote : for this Line tends the fame way towards which the infinite Leg is di- rected.
Reduction of Curves of the fecond Kind. Sir 7. Newton reduces all Curves of the fecond Kind to four Ciifes of Equation : In the firit, the Relation between the Ordinate and Abfciffe, making the Abfciffe *, and the Ordinate y, affumes this form xyy-[~ey — ax>-\-bxx-\~ cx~\-d. ~In the fecond Cafe, the Equation affumes this form xy = ax i -\-bx* -^~cx-\~d. In the third Cafe, the Equation is yy=a x z ~\-b x" -\-cx-\-d. In the fourth, the Equation is of this form, y=ax i -\-bx*-\~cx-\-d.
Enumeration of the Curves of the fecond Kind.
Under thefe four Cafes, the fame Author brings a vaft Number of different: Forms of Curves, to which he gives different Names.
A Hyperbola lying wholly in the Angle of the Afymp- totes, like a Conic Hvperbola, he calls an Infcribed Hyper- bola ; that which cuts the Afymptotes, and contains the Parts cut off within its own Periphery, a CircumfcriVd Hy- perbola ; that, one of whofe infinite Legs is inferib'd, the other circumfcrib'd, he calls Ambiginal ; that whofe Legs look towards each other, and are directed the fame way. Converging ; that where they look contrary ways, "Diverg- ing ; that where they are convex different ways, Crcfs-leg'd, that 1 applied to its Afymptote, with a concave Vertex, and diverging Legs, Conchoid cd -^ that which cuts its Afymptote with contrary Flexures, and is produced each way into con- trary Legs, Anguineous, or Snake-like 5 that which cuts its
Conjugate
