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for every different value of a there are n values of y, it will commonly happen that the ordinate will cut the curve in n or in n 2, n-4, &c, points, according as the equation has n, or n 2, n 4, &c, possible roots. It is not however to be inferred, that a right line will cut a curve of n dimensions, in n, n 4, &c, points, only; for if this were the case, a line of the 2d order, a conic section for instance, could, only be cut by a right line in two points ;-but this is manifestly incorrect, for though a conic parabola will be cut in two points by a right line oblique to the axis, yet a right line parallel to the axis can only cut the curve in one point.
29. It is true in general, that lines of the n order cannot be cut by a right line in more than n points; but it does not hence follow, that any right line whatever will cut in n points every line of that order; it may happen that the number of intersections is n − 1, n 2, n 3, &c, ton - n. The number of intersections that any right line whatever makes with a given curve line, cannot therefore determine the order to which a curve line appertains. For, as Euler remarks, if the number of intersections ben, it does not follow that the curve belongs to the n order, but it may be referred to some superior order; indeed it may happen that the curve is not algebraic, but transcendental. This case excepted, however, Euler contends that we may always affirm positively that a curve line which is cut by a right line in n points, cannot belong to an order of lines inferior to n. Thus, when a right line cuts a curve in 4 points, it is certain that the curve does not belong to either the second or third order of lines; but whether it be referred to the fourth, or a superior order, or whether it be transcendental, is not to be decided by analysis.
30. Dr. Waring has carried this enquiry a step further than Euler, and has demonstrated that there are curves of any number of odd orders, that cut a right line in 2, 4, 6, &c, points only; and of any number of even orders that cut a right line in 3, 5, 7, &c, points only; whence this author likewise infers, that the order of the curve cannot be announced from the number of points in which it cuts a right line. See his Proprietates Algebraicarum Curvarum.
31. Every geometrical curve being continued, either returns into itself, or goes on to an infinite distance. And if any plane curve has two infinite branches or legs, they join one another either at a finite, or at an infinite distance.
32. In any curve, if tangents be drawn to all points of the curve; and if they always cut the abscissa at a finite distance from its origin; that curve has an asymptote, otherwise not..
33. A line of any order may have as many asymptotes as it has dimensions, and no more.
34. An asymptote may intersect the curve in so many points abating two, as the equation of the curve has dimensions. Thus, in a conic section, which is the second order of lines, the asymptote does not cut the curve at all; in the third order it can only cut it in one point; in the fourth order, in two points; and so on.
35. If a curve have as many asymptotes, as it has dimensions, and a right line be drawn to cut them all, the parts of that measured from the asymptotes to the curve, will together be equal to the parts measured in the same direction, from the curve to the asymptotes.
36. If a curve of n dimensions have n asymptotes, then the content of the n abscissas will be to the content of the n ordinates, in the same ratio in the curve and asymptotes; the sum of their n subnormals, to ordinates perpendicular to their abscissas, will be equal to the curve and the asymptotes; and they will have the same central and diametral curves.
37. If two curves of n and m dimensions have a common asymptote; or the terms of the equations to the curves of the greatest dimensions have a common divisor; then the curves cannot intersect each other in n × m points, possible or impossible. If the two curves have a common general centre, and intersect each other in n x m points, then the sum of the affirmative abscissas, &c, to those points, will be equal to the sum of the negative; and the sum of the n subnormals to a curve which has a general centre, will be proportional to the distance from that centre.
38. Lines of the third, fifth, seventh, &c order, or any odd number, have, as before remarked, at least two infinite legs or branches, running contrary ways; while in lines of the second, fourth, sixth, or any even number of dimensions, the figure may return into itself, and be contained within certain limits.
39. If the right lines AP, PM, forming a given angle APM, cut a geometrical line of any order in as many points as it has dimensions, the product of the segments of the first terminated by P and the curve, will always be to the product of the segments of the latter, terminated by the same point and the curve, in an invariable ratio.
40. With respect to double, triple, quadruple, and other multiple points, or the points of intersection of 2, 3, 4, or more branches of a curve, their nature and number may be estimated by means of the following principles. 1. A curve of the n order is determinate when it is subjected to pass through
the number (~+1) · (~ + 2)
· 1 points. 2. A curve of the order cannot intersect a curve of the m order in more than min points.
Hence it follows that a curve of the second order, for example, can always pass through 5 given points (not in the same right line), and cannot meet a curve of the m order in more than mn points; and it is impossible that a curve of the m order should have 5 points whose degrees of multiplicity make together more than 2m points. Thus, a line of the fourth order cannot have four double points; because the line of the second order which would pass through these four double points, and through a fifth simple point of the curve of the fourth dimension, would meet 9 times; which is impossible, since there can only be an intersection 2 x 4 or 8 times.
For the same reason, a curve line of the fifth cannot, with one triple point, have more than three double points: and in a similar manner we may reason for curves of higher orders.
Again, from the known proposition, that we can always make a line of the third order pass through nine points, and that a curve of that order cannot meet a curve of the m order in more than 3m points, we may conclude that a curve of the m order cannot have nine points, the degrees of multiplicity of which make together a number greater than 3m. Thus, a line of the fifth order cannot have more than 6 double points; a line of the 6th order, which cannot have more than one quadruple point, cannot have with that quadruple point more than 6 double points; nor with two triple points more than 5 double points; nor even with one triple point more than 7 double points. Analogous conclusions obtain with respect to a line of the fourth order, which we may cause to pass through 14 points, and which can only meet a curve of the order in 4m points, and so on.
41. The properties of curves of a superior order, agree, under certain modifications, with those of all inferior orders. For though some line or lines become evanescent, and others become infinite, some coincide, others become equal; some points coincide, and others are removed to an infinite distance; yet, under these circumstances, the general properties still hold good with regard to the remaining quantities; so that whatever is demonstrated generally of any order, holds true in the inferior orders: and, on the contrary, there is hardly any property of the inferior orders, but there is some similar to it, in the superior ones.
For, as in the conic sections, if two parallel lines are drawn
te terminate at the section, the right line that bisects these will bisect all other lines parallel to them; and is therefore called a diameter of the figure, and the bisected lines ordinates, and the intersections of the diameter with the curve vertices; the common intersection of all the diameters the centre; and that diameter which is perpendicular to the ordinates, the vertex. So likewise in higher curves, if two parallel lines be drawn, each to cut the curve in the number of points that indicate the order of the curve; the right line that cuts these parallels so, that the sum of the parts on one side of the line, estimated to the curve, is equal to the sum of the parts on the other side, it will cut in the same manner all other lines parallel to them that meet the curve in the same number of points; in this case also the divided lines are called ordinates, the line so dividing them a diameter, the intersection of the diameter and the curve vertices; the common intersection of two or more diameters the centre; the diameter perpendicular to the ordinates, if there be any such, the axis; and when all the diameters concur in one point, that is the general centre.
Again, the conic hyperbola, being a line of the second order, has two asymptotes; so likewise, that of the third order may have three; that of the fourth, four; and so on: and they can have no more. And as the parts of any right line between the hyperbola and its asymptotes are equal; so likewise in the third order of lines, if any line be drawn cutting the curve and its asymptotes in three points; the sum of two parts of it falling the same way from the asymptotes to the curve, will be equal to the part falling the contrary way from the third asymptote to the curve; and so of higher curves.
Also, in the conic sections which are not parabolic: as the square of the ordinate, or the rectangle of the parts of it on each side of the diameter, is to the rectangle of the parts of the diameter, terminating at the vertices, in a constant ratio, viz, that of the latus rectum, to the transverse diameter. So in non-parabolic curves of the next superior order, the solid under the three ordinates, is to the solid under the three abscissas, or the distances to the three vertices; in a certain given ratio. In which ratio if there be taken three lines proportional to the three diameters, each to each; then each of these three lines may be called a laius rectum, and each of the corresponding diameters a transverse diameter. And, in the common, or Apollonian parabola, which has but one vertex for one diameter, the rectangle of the ordinates is equal to the rectangle of the abscissa and latus rectum : so, in those curves of the second kind, or lines of the third kind which
have only two vertices to the same diameter, the solid under the three ordinates, is equal to the solid under the two abscissas, and a given line, which may be reckoned the latus
Lastly, since in the conic sections where two parallel lines terminating at the curve both ways, are cut by two other parallels likewise terminated by the curve; we have the rectangle of the parts of one of the first, to the rectangle of the parts of one of the second lines, as the rectangle of the parts of the second of the former, to the rectangle of the parts of the second of the latter pair, passing also through the common point of their division. So, when four such lines are drawn in a curve of the second kind, and each meeting it in three points; the solid under the parts of the first line, will be to that under the parts of the third, as the solid under the parts of the second, to that under the parts of the fourth. And the analogy between curves of different orders may be carried much further: but as enough is given for the objects of this work; we shall now present a few of the most useful problems.
Knowing the Characteristic Property, or the Manner of Description of a Curve, to find its Equation.
This in most cases will be a matter of great simplicity; because the manner of description suggests the relation between the ordinates and their corresponding abscissas; and this relation, when expressed algebraically, is no other than the equation to the curve. Examples of this problem have already occurred in sec. 4 ch. i, of this volume: to which the following are now added to exercise the student.
Ex. 1. Find the equation to the cissoid of Diocles; whose manner of description is as below.
From any two points P, S, at equal distances from the extremities A, B, of the diameter of a semicircle, draw, ST, PM, perpendicular to AB. From the point T where ST cuts the semicircle, draw a right line AT, it will cut PM in M, a point of the curve required.
Now, by theor. 87 Geom. AS. SB ST2; and by the construction, As SB AP. PB. Also the similar triangles APM,
AST, give AP PM :: AS: ST :: PB : ST =