CURVES. 



figurcj one answering to another, then 

 these three right lines may be called the 

 latera recta of the figure, and the parts of 

 the diameter, between the vertices, the 

 latera transversa. And as in the conic 

 parabola, having to one and the same di- 

 ameter but one only vertex, the rectangle 

 under the ordinates is equal to that under 

 the part of the diameter cut off between 

 the ordinates and the vertex, and the la- 

 tus rectum -, so in curves of the second 

 order, which have but two vertices to the 

 same diameter, the parallelepiped under 

 three ordinates is equal to the parallele- 

 piped under the two parts of the diame- 

 ter, cut off between the ordinates and 

 those two vertices and a given right line, 

 which therefore may be called the latus 

 rectum. Moreover, as in the conic sec- 

 tions, when two parallels terminated on 

 each side of the curve are cut by two 

 other parallels terminated on each by 

 the curve, the first by the third, and 

 the second by the fourth ; as here the 

 rectangle under the parts of the first is to 

 the rectangle under the parts of the 

 third ; as the rectangle under the parts 

 of the second is to that under the parts 

 of the fourth ; so when four such right 

 lines occur in a curve of the second kind, 

 each in three points, then shall the paral- 

 lelepiped under the parts of the first right 

 line be to that under the parts of the 

 third, as the parallelepiped under the 

 parts of the second line to that under 

 the parts of the fourth. Lastly, the legs 

 of curves, both of the first, second, and 

 higher kinds, are either of the parabolic 

 or hyperbolic kind : an hyperbolic leg 

 being that which approaches infinitely 

 towards some asymptote ; a pai-abolic 

 that which has no asymptote. These 

 legs are best distinguished by their tan- 

 gents ; for if the point of contact go off 

 to an infinite distance, the tangent of the 

 hyperbolic leg will coincide with the 

 asymptote ; and that of the parabolic leg 

 recede infinitely and vanish. The as- 

 symptote, therefore, of any leg is found 

 by seeking the tangent of that leg to a 

 point infinitely distant, and the bearing 

 of an infinite leg is found by seeking the 

 position of a right line parallel to the tan- 

 gent, when the point of contact is infinite- 

 ly remote : for this line tends the same 

 way towards which the infinite leg is di- 

 rected. For the other properties of 

 curves of the second order, we refer the 

 reader to Mr. Maclaurin's treatise " De 

 Linearum geometricarum Proprietatibus 

 generalibus." 



Sir Isaac Newton reduces all curves of 

 the second order to the four following 



particular equations, still expressing theiri 

 all. In fhe first, the relation between 

 the ordinate and the abscisse, making the 

 abscisse x and the ordinate y, assumes 

 this form, x y*-\-e y = a x? + b x 2 -f- c x 

 -}-d. In the second case, the equation 

 takes this form, x y = a a-3 4- b x* -\-c x 

 -{-</ In the third case, the equation is 

 z/ l =a x3 4 b x* -{- c x 4- d. And in the 

 fourth case the equation rs of this form, 

 y = a a-3 4 b x 3 - 4 c x -|- d. Under 

 these four cases the same author enu- 

 merates seventy -two different forms of 

 curves, to which he gives different names, 

 as ambigenal, cuspidated, nodated, &c. 



CURVES, genesis of, of the second order 

 by shadoivs. If (says Sir Isaac Newton) 

 upon an infinite plane illuminated from a 

 lucid point the shadows of figures be pro- 

 jected, the shadows of the conic sections 

 will be always conic sections ; those of 

 the curves of the second kind will be al- 

 ways curves of the second kind ; those of 

 the curves of the third kind will be al- 

 ways curves of the third kind, and so on 

 in infinitum. And as a circle, by project- 

 ing its shadow, generates all the conic 

 sections, so the five diverging parabolas 

 by their shadows v/ill generate and ex- 

 hibit all the rest of the curves of the se- 

 cond kind ; and so some of the mostsim- 

 ple curves of the other kinds may be 

 found, which will form by their shadows 

 upon a plane, projecting from a lucid 

 point, all the rest of the curves of that 

 same kind. 



CURVES of the second order, having double 

 points. As curves of the second order 

 may be cut by a right line in three points > 

 and as two of these points are sometimes 

 coincident, these coincident intersections, 

 whether at a finite or an iniinite distance, 

 are called the double point. 



CURVES, use of, in the construction of 

 equations. One great use of curves in ge- 

 ometry is, by means of their intersec- 

 tions, to give the solution of problems. 



Suppose, ex gr. it were required to 

 construct the following equation of 9 di- 

 mensions. 



X9 4 b x t 4 ex 6 4- dxi 4- ex* + m -t-f.x? 

 4- g x* 4- h x 4- k = 0: assume the 

 equation to a cubic parabola x> = y ; 

 then, by writing y for x>, the given equa- 

 tion will become j/3 4- b x y 1 4 e y 1 4" d 

 x*y-\-exy+my -j-fx* + g ** -h h x 

 -|- k = ; an equation to another curve 

 of the second kind, where m or/ may be 

 assumed = 0, or any thing else : and by 

 the descriptions and intersections of these 

 curves will be given the roots of the 

 equation to be constructed. It is sujfn- 





