ASYMPTOTE

, is properly a right line, which approaches continually nearer and nearer to some curve, whose asympote it is said to be, in such sort, that when they are both indefinitely produced, they are nearer together than by any assignable finite distance; or it may be considered as a tangent to the curve when conceived to be produced to an infinite distance. Two curves are also said to be asymptotical, when they thus continually approach indefinitely to a coincidence: thus, two parabolas, placed with their axes in the same right line, are asymptotes to one another.

Of lines of the second kind, or curves of the sirst kind, that is the conic sections, only the hyperbola has asymptotes, which are two in number. All curves of the second kind have at least one asymptote; but they may have three. And all curves of the third kind may have four asymptotes. The conchoid, cissoid, and logarithmic curve, though not reputed geometrical curves, have each one asymptote. And the branch or leg of a curve that has an asymptote, is said to be of the hyperbolic kind.

The nature of asymptotes will be easily conceived from the instance of the asymptote to the conchoid. Thus, if, ABC &c be part of a conchoid, and the line MN be so drawn that the parts FB, GC, HD, IE, &c, of right lines, drawn from the pole P, be equal to each other; then will the line MN be the asymptote of the curve: because the perpendicular Cc is shorter than FB, and Dd than Cc, &c; so that the two lines continually approach; yet the points Ee &c can never coincide. |

Asymptotes of the Hyperbola are thus described. Suppose a right line DE drawn to touch the curve in any point A, and equal to the conjugate de of the diamete ACB drawn to that point A, viz, AD or AE equal to the semiconjugate Cd or Ce; then the two lines CDF, CEH, drawn from the centre C, through the points D and E, are the two asymptotes of the curve.

The parts of any right line, lying between the curve of the common hyperbola and its asymptotes, are equal to one another on both sides, that is, gG=hH. In like manner, in hyperbolas of the second kind, if there be drawn any right line cutting both the curve and its three asymptotes in three points, the sum of the two parts of that right line extended in the same direction from any two of the asymptotes to two points of the curve, is equal to the third part which extends in the contrary direction from the third asymptote to the third point of the curve.

If AGK be an hyperbola of any kind, whose nature, with regard to the curve and asymptote, is expressed by this general equation x^myⁿ = a^m4n, where x is=CF, and y=FG drawn any where parallel to the other asymptote CH; and the parallelogram CFGI be completed: Then m—n is to n, as this parallelogram CFGI is to the hyperbolic space FGK, contained under the determinate line FG, with the asymptote FK and the curve GK, both indefinitely continued towards K. So that, if m be greater than n, the said asymptotic space is finite and quadrable: but when m=n, as in the common or conic hyperbola, then m-n=o, the ratio of that space to the said parallelogram, is as n to o: that is, the hyperbolic space is insinitely great, in respect of the finite parallelogram: and when m is less than n, then, m-n being negative, the asymptotic space is to the determinate parallelogram, as a positive number is to a negative one, and is what Dr. Wallis calls more than infinite.

Asymptote of the Logarithmic Curve. If LMN be the logarithmic curve, QON an asymptote, LQ and MP ordinates, MO a tangent, and PO the subtangent, which in this curve is a constant quantity. Then the indeterminate space LMNQ is equal to LQ X PO, the rectangle under the ordinate LQ and the constant subtangent PO; and the solid generated by the rotation of that curve space about the asymptote NQ, is equal to half the cylinder, whose altitude is the said constant subtangent PO, and the radius of its base is LQ.

, by some, are distinguished into various orders. The asymptote is said to be of the first order, when it coincides with the base of the curvilinear figure: of the 2d order, when it is a right line parallel to the base: of the 3d order, when it is a right line ob- lique to the base: of the 4th order, when it is the common parabola, having its axis perpendicular to the base: and, in general, of the n + 2 order, when it is a parabola whose ordinate is always as the n power of the base. The asymptote is oblique to the base, when the ratio of the first fluxion of the ordinate to the fluxion of the base, approaches to an assignable ratio, as its limit; but it is parallel to the base, or coincides with it, when this limit is not assignable.

The doctrine and determination of the asymptotes of curves, is a curious part of the higher geometry. Fontenelle has given several theorems relating to this subject, in his Geometrie de l'Infini. See also Stirling's Lineæ Tertii Ordinis, prop. vi, where the subject of Asymptotes is learnedly treated; and Cramer's Introduction à l'analyse des lignes courbes, art. 147 & seq. for an excellent theory of asymptotes of geometrical curves and their branches. Likewise Maclaurin's Algebra, and his Fluxions, book i, chap. 10, where he has carefully avoided the modern paradoxes concerning infinites and infinitesimals. But the easiest way of determining asymptotes, it seems, is by considering them as tangents to the curves at an infinite distance from the beginning of the absciss, that is when the absciss x is infinite in the equation of the curve, and in the proportion of x^. to y^., or in that of the subtangent to the ordinate.

The areas bounded by curves and their asymptotes, though indefinitely extended, have sometimes limits to which they may approach indefinitely near: and this happens in hyperbolas of all kinds, except the sirst or Apollonian, and in the logarithmic curve; as was observed above. But in the common hyperbola, and many other curves, the asymptotical area has no such limit, but is infinitely great.—Solids, too, generated by hyperbolic areas, revolving about their asymptotes, have sometimes their limits; and sometimes they may be produced till they exceed any given solid.—Also the surface of such solid, when supposed to be infinitely produced, is either finite or infinite, according as the area of the generating figure is finite or infinite.