CURVE. 



intimate contact between the curve and 

 the circle of curvature ; and that a conic 

 section may be described, that shall have 

 the same curvature with a given line at a 

 given point, and the same variation of a 

 curvature, or a contact of the same kind 

 with the circle of curvature. The rays 

 of curvature of similar arches, in similar 

 figures, are in the same ratio as any ho- 

 mologous lines'of these figures, and the 

 variation of curvature is the same. See 

 CUKVE. 



CURVE, in geometry, a line, which, 

 running on continually in all directions, 

 may be cut by one right line in more 

 points than one. Curves are divided into 

 algebraical or geometrical, and transcen- 

 dental. Geometrical or algebraical curves 

 are those, whose ordiriates and abscisses 

 being right lines, the nature thereof can 

 he expressed by a finite equation having 

 those ordinates and abscisses in it. 



Transcendental curve, is such as, when 

 expressed by an equation, one of the 

 terms thereof is a variable quantity. 



Geometrical lines or curves are diVid- 

 ed into orders, according to the number 

 of dimensions of the equation expressing 

 the relation between the ordiuates and 

 abscisses, or according to the number of 

 points by which they may be cut by a 

 right line. So that a line of the first or- 

 der will be only a right line, expressed 

 by the equation y-\-ax-\-d = 0. A line 

 of the second, or quadratic order, will be 

 the conic sections and circle, whose most 

 general equation is?/ 1 -f ax -f b X y + c 

 x 1 -}- d x + e = 0. A line of the third 

 order, is that whose equations has 

 three dimensions, or may be cut by a 

 right line in three points, whose most 

 general equation is #3 _j_ a x 4- 6 x y 1 -}- 

 tx 1 -j- dx + e X y +/* 3 +&X 1 + hx+ 

 k 0. A line of the fourth order, is that 

 whose equation has four dimensions, or 

 which may be cut in four points by a 

 right line, whose most general equation 

 a x+b X y3-f ex* -f- dx -f e X y l 



+ ^ 3 + te + k X y -f te* nix* -f 

 n oc 1 -f- p x -f- q = 0, And so on. 



And a curve of the first kind (for aright 

 line is not to be reckoned among curves) 

 is the same with a line of the second or- 

 der ; and a curve of the second order, 

 the same as a line of the third ; and a line 

 of an infinite order, is that which a right 

 line can cut in an infinite number of 

 points, such as a spiral, quadratrix, cy- 

 cloid, the figures of the sines, tangents, 

 secants, and every line which is eene- 



TOL, IV. 





rated by the infinite revolutions of 

 cle or wheel. 



As to the curves of the second order, 

 Sir Isaac Newton observes they have 

 parts and properties similar to those of 

 the Hrst. Thus, as the conic sections have 

 diameters and axes, the lines cut by these 

 are called ordinates, and the intersection 

 of the curve and diameter, the vertex ; 

 so in curves of the second order, any two 

 parallel lines being drawn so as to meet 

 the curve in three points, a right line cut- 

 ting these parallels, so as that the sum of 

 the two parts between the secant and the 

 curve on one side is equal to the third 

 part terminated by the curve on the other 

 side, will cut in the same manner all other 

 right lines parallel to these, and meet the 

 curve in three parts, so as that the sum of 

 the two parts on one side will be still 

 equal to the third part on the other side. 

 These three parts, therefore, thus 

 equal, may be called ordinates or appli- 

 cates : the secant may be styled the dia- 

 meter ; the intersection of the diameter 

 and the curve the vertex ; and the point 

 of concourse of any two diameters the 

 centre. And if the diameter be normal 

 to the ordinates, it may be called axis 

 and that point where all the diameters 

 terminate the general centre. Again, as 

 an hyperbola of the first order has two 

 asymptotes; that of the second three; 

 that of the third four, &c. : and as the 

 parts of any right line, lying between the 

 conic hyperbola and its two asymptotes, 

 are every where equal ; so in the hyper- 

 bola of the second order, if any right line 

 be drawn, cutting both the curve and its 

 three asymptotes in three points, the 

 sum of the two parts of that right line, be- 

 ing drawn the same way from any two 

 asymptotes to two points of the curve, 

 will be equal to a third part drawn a con- 

 trary way from the third asymptote to a 

 third point of the curve. Again, as in 

 conic sections not parabolical, the square 

 of the ordinate, that is, the rectangle un- 

 der the ordinates, drawn to contrary sides 

 of the diameter, is to the rectangle of the 

 parts of the diameter which are termi- 

 nated at the vertices of the ellipsis or hy- 

 perbola, as the latus rectum is to the latus 

 transversum ; so in non-parabolic curves 

 of the second order, a parallelepiped un- 

 der the three ordinates is to a parallele- 

 piped under the parts of the diameter, 

 terminated at the ordinates, and the three 

 vertices of the figure, in a certain given 

 ratio ; in which ratio, if you take three 

 right lines, situated at the three parts of 

 the diameter between the vertices of the 



