CUR ( 3<$
r niueate a-crofs, Cruciform ; that which returning around
b it fclf. Nodated; that whofe two Parts concur in the ? 2 le of Contact, and there terminate, Cufpidated; that
hofe Conjugate is oval, and infinitely (mall, i. e. a Point, (pointed ; and that which from the Impoflibility of its two n o'ts ' s ' w' tnout e ' tner Oval, Node, Cufp, or Point pure : And i° f ^ e ^ ame manncr h e denominates a Parabola, to be
.merging, diverging, cruciform, &c. "Where the Number
c Hyperbolic Legs, exceeds that of the Conic Hyperbola ; ? ,j en ominates the Hyperbola redundant.
jfow, the various Curves which he enumerates under thefe four Cafes, are in Number 72. ; whereof nine are re- Jutidtmt Hyperbolas, without Diameters, having three A- Liptotes including a Triangle.
' The firft confiding of three Hyperbolas, one inferibed, another ciroumfcribed, another ambigonal, with an Oval ;
1,5 fecond Nodated ; the third Cuffidated ; the fourth f tinted ; the fifth and fixth 'Pure ; the feventh and eighth unciform ; the laft Anguineal.
Thefe are 12 redundant Hyperbolas,, having only one Diameter : The firft Oval, the fecond Nodated, the third Cuffidated, the fourth 'Pointed, the fifth, fixth, feventh, a n/ eighth, 'Pure ; the ninth and tenth Cruciform 5 the eleventh and twelfth Conchoidal.
Two are redundant Hyperbolas with three Diameters.
jsjine are redundant Hyperbolas, with three Afymptotes converging to a common Point ; the fitft form'd of the jifth and fixth redundant Patabolas, whofe Afymptotes in- clude a Triangle ; the fecond, of the feventh and eighth ; the third and fourth, of the ninth ; the fifth is form'd of the fifth and feventh of the redundant Hyperbolas, with one Diameter} the fixth, of the fixth and feventh 5 the feventh, of the eighth and ninth ; the eighth, of the tenth and ele- venth ; the ninth, of the twelfth and thirteenth : All which Converfions are effected, by diftinguifhing the Triangle comprehended between the Afymptotes, till it vanifh into a Point.
Six are defective Parabolas, having no Diameters : The firft Oval, the fecond Nodated, the third Cufpidated, the fourth Pointed, the fifth 'Pure.
Seven are defective Hyperbolas, having Diameters : The firft and fecond Conchoidal, with an Oval ; the third Noda- tii, the fourth Cufpidated, which is the Ciflbid of the An- tients j the fifth and fixth, 'Pointed ; the feventh 'Pure.
Seven are Parabolic Hyperbolas, having Diameters : The firft Oval, the fecond Nodated, the third Cufpidated, the fourth Pointed, the fifth 'Pure, the fixth Cruciform, the feventh Anguineous.
Four are Parabolic Hyperbolas. Four are Hyperbolifms of the Hyperbola. Three Hyperbolas of the Ellipfis. Two Hyperbolifms of the Parabola.
Five are diverging Parabolas : one, a 'trident ; the fe- cond Oval, the third Nodated, the fourth 'Pointed, the fifth Cuffidated ; (this is Neal's 'Parabola, ufually called the Se- meubic Parabola :) the fixth, 'Pure.
Laftly, one commonly call'd the Cubic 'Parabola;
Organical Sefcripion of thefe Corves.
ift, If two Angles given in Magnitude, PAD, PBD, (Tah. Analyjis, Fig. 53.) revolve round Poles given in pofition, A and B 3 and their Legs, A P, B P, with their Point of Concourfe, P, pafs over another right Line : The other two Legs AD, BD, with their Point of Concoutfe D, will defcribe a Conic Section palling thro' the Poles A B : Unlefs that Line happen to pafs thro' either of the Poles A or B ; or unlefs the Angles BAD and A B D vanifh to- gether : in which Cafes, the Point will defcribe a right Line.
idly, Now, if the Legs, A P, B P, by their Point of Con- courfe, P, thus defcribe a Conic Section paffing thro' one of ™ Poles, A ; the other two, A D, B D, with their Point of Concoutfe D, will defcribe a Carve of the fecond Kind, faffing thro' the other Pole B, and having a double Point in •he firft Pole A : Unlefs the Angle BAD, A B D, vanifh together ; in which Cafe, the Point D will defcribe another <-°mc Seffion, pafling thro' the Pole A.
jdly, If the Conic SeBion defcrib'd by the Point P, pafs thro' neither of the Poles A B ; the Point D will defcribe
- V\ e "f 'bufitfnd or third Kind, having a double Point :
- ">«> double Point will be found in the Concourfe of the
menbing Legs AD, BD: When the two Angles BAP, v or, vanifh together, the Curve defcribed will be of the J'cmtd Kiad, when the Angles BAD, ABD, vanifh toge- "«; otherwife of the third Kind, having two other double
UrV" tho Polcs A and B - . with j e g ar[ j to double 'Points of Curves : We haveobfetv'd
GUM.
that Car ve s of the fecond Kind may be cut by a right • , ne ln thofe Points : Now two of thefe fometimes coin-
f % ri ^' w ' ien t ' ie right Line paffes thro' an infinitely "a" Oval ; or thro' the Concourfe of two Parts of a Curve,
tim " a I cu f tin 8 eacn - othe '. and uniting in a Cufp. Some- « the right Line, even, only cuts the Curve in one
?l nt L aS ," °^u^ eS r. 0f the C V ef,m and Cubic Vari- bola, &c In which Cafe, we muft conceive the right Lines pafltng thro two other Points of a Curve, placed as it were, at an infinite diftance : Now thefe coincident Interfec- tions, whether at a finite or infinite Diftance, make what we call double Point.
Genefis of Curves of the fecond Order by Shadows.
If the Shadows of Figures be projected on an infinite Plane, illumin'd by a lucid Point ; the Shadows of Conic Seflions will ftm be Conic Sections ; thofe of Curves of the fecond Kind, will be Curves of the fecond Kind ; thofe of the third Kind, Curves of the third Kind, £S?c.
And as a Curve, in projecting a Shadow, generates all the Conic Sections ; fo, the five diverging Parabolas, with their Shadows, generate and exhibit all other Curves of the fe- cond Kind.
And in this manner may a Train of the fimple Curves of other Kinds be found, which fhall form all the other Curves of the fame Kind, by their Shadows projeSed from a lucid Point, upon a Plane.
Defcription of Corves of the fecond Order, having double 'Points.
Thefe are all defcrib'd from feven given Points, whereof one is the double Point it felf : Thus, let thete be given any feven Points of the Curve to be defcribed ; as, v.g. A, B,C,D, E,F, G, (Tab. Analyfis, Fig. 54.) wheteof A is the double Point : join the Point A, and any other two Points, v. g. B and C ; and let the Angle C A B of the Tri- angle ABC, revolve about its Vertex A ; and another of the Angles ABC, about its Vertex B. And when the Point of Concourfe C, of the Legs A C, B C, is fuccefiively applied to the four other Points D,E,F,G, let the Concoutfe of the remaining Legs AB and B A, fall on the four Points P, Q_, R, S.
Thro' thofe four Points, and the fifth A, defcribe a Conic Section ; and let the foremention'd Angles CAB, C B A, fo revolve, as that the Point of Concourfe of the Legs A B, B A, may pafs over that Conic Section ; and the Concoutfe of the other Legs AC, BC, will defcribe the propofed Curve.
Ufe of thefe Corves in the Conjlruttion of Equations:
The ufe of Curves in Geometry, is by means of the In- terferons thereof, to folve Problems. See Construction.
Suppofe, v. g. an Equation to be conflructed of nine Di- menfions, as x$-\-bx* -\- ex 6 -\-dx* -\-ex*~[-fx* ~\~gx' -\-hx-\-r = o; where b, c, d, ckc. fignify any given Quan- tities affected with the Signs -f- and — : aflume an Equation to a Cubic Parabola x' =y ■ and the firft Equation, wri- ting y for x* will come out y* ~\-b xy* -^- c y* -\-dx'y-L- exy--my-\-f x' --gx'-\-hx-\-k=o ; an Equation to another Curve of the fecond Kind, where m ox f may be affum'd or annulled. And by the Defcriptions and Inrerfec- tions of thefe Curves will be given the Roots of the Equa- tion to be conftructed. 'Tis fufficient to defcribe the Cubi- cal Parabola once.
If the Equation to be conftructed, by omitting the two laft Terms h x and k, be reduced to feven Dimenfions ; the other Curve, by expunging m, will have the double Point in the beginning of the Abfcifle, and may be eafily de- fcrib'd as above : If it be reduced to fix Dimenfions, by omitting the three' laft, taking g x' -\- h x -J- k ; the other Curve, by expunging /, will become a Conic Section : and if, by omitting the three laft Terms, the Equation be re- duced to three Dimenfions, we fhall fall on Dr. Wallis'f* Con- ftruction by the Cubic Parabola and right Line.
Rectification of a Curve, is the finding of a right Line equal to a Curve. For the 'Praxis hereof, fee Rectifi- cation of Curves.
Quadrature of a Curve, the finding of the Area, or Space included by a Curve ; or the afligning of a Square equal to a Curvilinear Space. See Quadrature.
Family of Curves, is an Aflemblage of feveral Curves of different Kinds, all defined by the fame Equation of an indeterminate Degree ; but differently, according to the Diveifity of Kind.
E. g. Suppofe an Equation of an indeterminate Degree, a m — 1 x =y m . If m = z, then will a x — y 1 ; if m^= 3, then will «'* = y' i if (0 = 4, then a' x=:y" y g?c. All which Curves are faid to be of the fame Family, or tribe.
The Equations whereby the Families of Curves are defined, are not to be confounded with the Tranfcendental ones : For tho, with regard to the whole Family, they be of an indeterminate degree, yet, with refpect to each feveral Curve out of the Family, they are determinate ; whereas tranfeendent Equations are of an indefinite Degree, with, refpect to the fame Curve.
All Algebraic Curves, therefore, compofe a certain Fa- mily, confifting of innumerable others; each whereof com- prehends infinite Kinds. For fince the Equations whereby the Curves are defined enter the FaEia, either of the Powers A a a a a of
