equation, as if the circle together with the right line formed only one continued curve.

10. This difference between simple and complex curves being once established, it is manifest that the lines of the second order are either continued curves, or complex lines formed of two right lines; for if the general equation have rational factors, they must be of the first order, and conse. quently will denote right lines. Lines of the third order will be either simple, or complex, formed either of a right line and a line of the second order, or of three right lines. In like manner, line of the fourth order will be continued and simple, or complex, comprising a right line and a line of the third order, or two lines of the second order, or lastly, four right lines. Complex lines of the fifth and superior orders will be susceptible of an analogous combination, and of a similar enumeration. Hence it follows, that any order whatever of lines may comprise, at once, all the lines of inferior order, that is to say, that they may contain a complex line of any inferior orders with one or more right lines, or with lines of the second, third, &c. order; so that if we sum the numbers of each order, appertaining to the simple lines, there will result the number indicating the order of the complex line.

Def. 9. That is called an hyperbolic leg, or branch of a curve, which approaches constantly to some asymptote; and that a parabolic one which has no asymptote.

ART. 11. All the legs of curves of the second and higher kinds, as well as of the first, infinitely drawn out, will be of either the hyperbolic or the parabolic kind: and these legs are best known from the tangents. For if the point of contact be at an infinite distance, the tangent of a hyperbolic leg will coincide with the asymptote, and the tangent of a para. bolic leg will recede in infinitum, will vanish and be nowhere found. Therefore the asymptote of any leg is found by seeking the tangent to that leg at a point infinitely distant and the course, or way of an infinite leg, is found by seeking the position of any right line which is parallel to the tangent where the point of contact goes off in infinitum; for this right line is directed the same way with the infinite leg.

Sir Isaac Newton's reduction of all lines of the third order, to four cases of equations; with the enumeration of those lines.

CASE I.

12. All the lines of the first, third, fifth, and seventh order, or of any odd order, have at least two legs or sides proceeding on ad infinitum, and towards contrary parts. And all lines

of the third order have two such legs or branches running out contrary ways, and towards which no other of their infinite legs (except in the Cartesian parabola) tend. If the legs are of the hyperbolic kind, let Gas be their asymptote; and to it GE

let the parallel CBC be drawn, terminated (if possible) at both ends at the curve. Let this parallel be bisected in x, and then will the locus of that point x be the conical or common hyperbola xq, one of whose asymptotes is as. Let its other asymptote be Ab. Then the equation by which the relation between the ordinate BC=y, and the abscissa AB = x, is determined, will always be of this form: viz.

xy2 + ey = ax2 + bx2 + cx + d . . . (I.)

Here the coefficients e, a, b, c, d, denote given quantities, affected with their signs + and, of which terms any one may be wanting, provided the figure through their defect does not become transformed into a conic section. The conical hyperbola xe may coincide with its asymptotes, that is, the point x may come to be in the line AB; and then the term +ey will be wanting.

CASE II.

13. But if the right line CBC cannot be terminated both ways at the curve, but will come to it only in one point; then draw any line in a given position which shall cut the asymp tote as in a ; as also any other right line, as BC, parallel to the asymptote, and meeting the curve in the point c; then the equation, by which the relation between the ordinate BC and the abscissa AB is determined, will always assume this form: viz. xy = ax3+ bx2+ cx + d. (II.) VOL. II.

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CASE III.

14. If the opposite-legs be of the parabolic kind, draw the right line CBC, terminated at both ends (if possible) at the curve, and running according to the course of the legs; which line bisect in B: then shall the locus of в be a right line. Let that right line be AB, terminated at any given point, as a : then the equation, by which the relation between the ordinate BC and the abscissa AB is determined, will always be of this form: yax3 + bx2 + cx + d . . . . (III.)

CASE IV.

15. If the right line CBC meet the curve only in one point, and therefore cannot be terminated at the curve at both ends: let the point where it comes to the curve be c, and let that right line at the point B, fall on any other right line given in position, as AB, and terminated at any given point, as a. Then will the equation expressing the relation between BC and AB, assume this form :

16. In the first case, or that of equation 1, if the term ax3 be affirmative, the figure will be a triple hyperbola with six hyperbolic legs, which will run on infinitely by the three asymptotes, of which none are parallel, two legs towards each asymptote, and towards contrary parts; and these asymp totes, if the term br2 be not wanting in the equation, will mutually intersect each other in 3 points, forming thereby the triangle Ddd. But if the term br2 be wanting, they will all con- verge to the same point. This kind of hyperbola is called redundant, because it exceeds the conic hyperbola in the number of its hyperbolic legs.

In every redundant hyperbola, if neither the term ey be wanting, nor b2 - 4ac = ae √ a, the curve will have no diameter but if either of those occur separately, it will have only one diameter; and three, if they both happen. Such diameter will always pass through the intersection of two of the asymptotes, and bisect all right lines which are terminated each way by those asymptotes, and which are parallel to the third asymptote.

=

17. If the redundant hyperbola have no diameter, let the four roots or values of x in the equation axa1 + bx3 + cx2 + dx + e2 O, be sought; and suppose them to be AP, AW, AT, and Ap (see the preceding figure). Let the ordinates PT, τ, 71, pl, be erected; they shall touch the curve in the points, T, 7, 7, t, and by that contact shall give the limits of the curve, by which its species will be discovered.

Thus, if all the roots AP, A☎, AT, ap, be real, and have the same sign, and are unequal, the curve will consist of three hyperbolas and an oval: viz. an inscribed hyperbola, as Ec; a circumscribed hyperbola, as roc; an ambigeneal hyperbola, (i. e. lying within one asymptote and beyond another) as pt; and an oval 7. This is reckoned the first species. Other relations of the roots of the equation, give 8 more different species of redundant hyperbolas without diameters; 12 each with but one diameter; 2 each with three diameters; and 9 each with three asymptotes converging to a common point. Some of these have ovals, some points of decussation, and in some the ovals degenerate into nodes or knots.

18. When the term ar3 in equa. 1, is negative, the figure expressed by that equation will be a deficient or defective hyperbola; that is, it will have fewer legs than the complete conic hyperbola. Such is the marginal figure, representing Newton's 33d species; which is constituted of an anguineal or serpentine hyperbola (both legs approaching a common asymptote by means of a contrary flexure), and a conjugate oval. There are 6 species of de. fective hyperbolas, each having but one asymptote, and only two hyperbolic legs, running out contrary ways, ad

π

T

APP

infinitum; the asymptote being the first and principal or dinate. When the term ey is not absent, the figure will have no diameter; when it is absent, the figure will have one diameter. Of this latter class there are 7 different species, one of which, namely, Newton's 40th species, is exhibited in the margin.

19. If, in equation 1, the term ar3 be wanting, but br2 not, the figure express. ed by the equation remaining, will be a parabolic hyperbola, having two hyperbolic legs to one asymptote, and two pa. rabolic legs converging one and the same way. When the term ey is not wanting, the figure will have no diamemeter; if that term be wanting, the figure will have one diameter. There

A 2

are 7 species appertaining to the former case; and 4 to the

latter.

20. When, in equa. 1, the terms ar3, br3, are wanting, or when that equation becomes xy' + ey = cx + d, it expresses a figure consisting of three hyperbolas opposite to one an other, one lying between the parallel asymptotes, and the

bther two without: each of these curves having three asymp totes, one of which is the first and principal ordinate, the other two parallel to the abscissa, and equally distant from it; as in the annexed figure of Newton's 60th species. Otherwise the said equation expresses two opposite circumscribed hyperbolas, and an anguineal hyper. bola between the asymptotes. Under this class there are 4 species, called

d

A

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by Newton Hyperbolisma of an hyperbola. By hyperbolismæ of a figure he means to signify when the ordinate comes out, by dividing the rectangle under the ordinate of a given conic section and a given right line, by the common abscissa.

21. When the term ca2 is negative, the figure expressed by the equation xy3+ey=cx+d, is either a serpentine hyperbola, having only one asymptcie, being the principal ordinate; or else it is a conchoidal figure. Under this class there are three species, called Hyperbolismæ of an ellipse.

22. When the term ca2 is absent, the equa. xy2 + ey = á,. expresses two hyperbolas, lying, not in the opposite angles of the asymptotes (as in the conic hyperbola), but in the adjacent angles. Here there are only 2 species, one consisting of an inscribed and an ambigeneal hyperbola, the other of two inscribed hyperbolas. These two species are called the Hyperbolisma of a parabola.

23. In the second case of equations, or that of equation II, there is but one figure; which has four infinite legs. Of these, two are hyperbolic about one asymptote, tending towards contrary parts, and two converging parabolic legs, making with the former nearly the figure of a trident, the familiar name given to this species. This is the Cartesian parabola, by which equations of 6 dimensions are sometimes constructed it is the 66th species of Newton's enumeration. 24. The third case of equations, or equa. III, expresses a figure having two parabolic legs running out contrary ways of these there are 5 different species, called diverging or bellform parabolas; of which 2 have ovals, 1 is nodate, 1 punctate, and 1 cuspidate. The figure shows Newton's 67th species; in which the oval must always be so small that no right line which cuts it twice can cut the parabolic curve ct more than once.

G

A

B