Hyperbola

“ In the hyperbola we have the mathematical demonstration of the error of an axiom. In the infinite inch we behold an absurdity mathematically demonstrated. So that it appears we can give mathematical demonstration in support of untruth, impossibilities and absurdities; and our reason can not discover the error of the reasoning! Alas, for poor humanity, if an endless destiny depended upon such scientific certainty! Yet mathematical reasoning about abstract truth is universally conceded to be less liable to error than any other form of scientific analysis. ”

“ The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. ”

“ Suppose, indefinitely, the line AGK to be an hyperbola described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G and its asymptote in V, VG may be reciprocally as any power DNn of the line ZX or DN, whose index is the number n: to find the density of the medium in which a projected body will describe this curve. ”

“ For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola.
The case when three points are given is more readily solved thus. Let B, C, D, be the given points. Join BC, CD, and produce them to E, F, so as EB may be to EC as SB to SC; and FC to FD as SC to SD. On EF drawn and produced let fall the perpendiculars SG, BH, and in GS produced indefinitely take GA to AS, and Ga to aS, as HB is to BS; then A will be the vertex, and Aa the principal axis of the trajectory ”

“ The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.
In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus. ”

“ Fourthly, we may propose to ourselves this problem: suppose the planet to be once put in motion, and then continually attracted by the sun with a force inversely as the square of the distance from the sun, what curve will it describe? It is found by the investigation which I have spoken of, that the curve which it will describe will be one or other of the following a circle, an ellipse, a parabola, or a hyperbola; and that the sun will be in the centre of the circle, or in the focus of the ellipse, parabola, or hyperbola. In nature we do not know any instance of the hyperbola ”

“ Contains 10 Propositions, from 41 to 51; on the 42th whereof he finds a Sphere equal to an Hyperbolical Ring-solid; whence divers ways are open'd towards the attaining the Quadrature of the Hyperbola: And he finds at Sphere equal to a Ring made by the Rotation of a Segment of an Hyperbola, and of the Segment of a Circle thereto annexed, dcscribed about the Base of the Hyperbola as a Chord Line: Then he absolutely cubes certain Hoofs cut out of an Hyperbolical Cylinder, and thence derives other ways towards the obtaining the Quadrature of the Hyperbola. ”

“ Now here we have four different species of curve—circle, ellipse, parabola, and hyperbola—each having its peculiar properties and its separate equation, and the first and last of which are quite opposite in nature, connected together as members of one series, all producible by a single process of insensible modification.
But the blindness of those who think it absurd to suppose that complex organic forms may have arisen by successive modifications out of simple ones, becomes astonishing when we remember that complex organic forms are daily being thus produced. ”

“ First, if the three intersections by the line infinity are all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the defective hyperbolas, with one hyperbolic branch. Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas ”

“ In the ellipse, which lies between the directrices, d1 + d2 is constant, therefore also r1 + r2. In the hyperbola on the other hand d1 − d2 is constant, equal to the distance between the directrices, therefore in this case r1 − r2 is constant.
If we call the distances of a point on a conic from the focus its focal distances we have the theorem:
In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the focal distances is constant. ”

“ In determining by the analysis of probabilities the ratio of the chances which, in these limits, give an appreciable hyperbola, to the chances which give an orbit which may be confounded with a parabola, I have found that it is a bet of at least 6000 against one that a nebula which penetrates into the activity of the sun in such a manner as to be observed will describe either a very elongated ellipse or an hyperbola. ”

“ As if the ordinate describe a parabola, m being = 2, and n = 1, the force will be as the given quantity 2B°, and therefore is given. Therefore with a given force the body will move in a parabola, as Galileo has demonstrated. If the ordinate describe an hyperbola, m being = 0 - 1, and n = 1, the force will be as 2A-3 or 2B3; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. ”

“ Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the area of the hyperbola [228] which led to Napier's logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one has ever squared the circle with so much genius, or, excepting his principal object, with so much success. ”

“ When the conic is a hyperbola the meridian line is in the form of a looped curve (fig. 12) . The corresponding figure of the film is called the nodoid. The resultant of the internal pressure and the surface-tension is equivalent to a pressure along the axis equal to that due to a pressure p acting on a circle whose diameter is the conjugate axis of the hyperbola.
When the conjugate axis of the hyperbola is made smaller and smaller, the nodoid approximates more and more to the series of spheres touching each other along the axis. ”

“ Again: it is mathematically demonstrable, that a straight line, the asymptote of a hyperbola, may eternally approach the curve of the hyperbola and never meet it. But no axiom can be plainer than that if two lines continually approach each other they must at length meet. Here is a demonstration contradicting an axiom; and no man has ever yet shown the possibilities of reconciling them, nor yet of denying either side of the contradiction.
Again: it is a fundamental axiom, contained in the definition of a circle, that it must have a center ”

“ The general equations expressing the motion of a planet considered as a material particle round a centre of attraction lead to theorems the more interesting of which will now be enunciated.
(1) The motion of such a planet may take place not only in an ellipse but in any curve of the second order; an ellipse, hyperbola, or parabola, the latter being the bounding curve between the other two. A body moving in a parabola or hyperbola would recede indefinitely from its centre of motion and never return to it. The ellipse is therefore the only closed orbit. ”

“ Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches. A branch is either re-entrant, or it extends both ways to infinity, and in this case, we may regard it as consisting of two legs (crura, Newton) , each extending one way to infinity, but without any definite separation. ”

“ Rectang′ūlarly, with, or at, right angles.—n. Rectang′ūlarness.—Rectangular hyperbola, a hyperbola whose asymptotes are at right angles to one another; Rectangular solid, a solid whose axis is perpendicular to its base. ”

“ The equations of this typical transformation may easily be derived by considering that the equation of the rectangular hyperbola
(in the plane of the variables x, ct) is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters. The equations of transformation are thus found to be
where a denotes a constant. ”

“ Given the curve representing attraction y = 1 / x2 the machine draws the hyperbola y = 1 / x the curve representing potential, as the work done in bringing a unit from an infinite distance to a point is measured by the area between the curve of attraction, the axis of x, and the ordinate at that point. ”

“ Inasmuch as pm - ph = a constant quantity. If we have the time given at which the shock or wave arrived at a third station as at Iquique, it is evident that a second hyperbola r′ v′ t′ might be drawn with Iquique and Huanillos as foci, and that the mutual intersection of these two hyperbolas with a third hyperbola, having for its foci Iquique and Mejillones, would give the origin of the wave. 206 The obtaining of a mutual intersection would depend on the assumed velocity, and the accuracy of the result, like that of the method of circles, would depend upon the trials we made. ”

“ Asymptotes of the hyperbola.—The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.—The portions of a secant comprised between the hyperbola and its asymptotes are equal.—Application to the tangent and to its construction.
The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel. ”

“ Thus, among plane curves, if a point moves so as to remain constantly at the same distance from a fixed point, it will generate a circle; if it is the sum or the difference of its distances from two fixed points which remains constant, the curve described will be an ellipse or an hyperbola; if it is their product, we shall have an entirely different curve; if the point departs equally from a fixed point and from a fixed line, it will describe a parabola; if it revolves on a circle at the same time that this circle rolls along a straight line, we shall have a cycloid ”

“ From the line-pair we pass through an infinity of hyperbolas to the parabola, and thence through an infinity of ellipses to the circle. Related to the sections are certain remarkable points which have no name. Kepler calls them foci. The circle has one focus at the centre, an ellipse or hyperbola two foci equidistant from the centre. The parabola has one focus within it, and another, the “caecus focus,” which may be imagined to be at infinity on the axis within or without the curve. The line from it to any point of the section is parallel to the axis. ”

“ Fig. 129.—An Ellipse, Parabola and Hyperbola.
Second, they shine chiefly with their own light like the stars; third, they do not travel in small circles around the Sun like the planets; but the paths they take are either long ovals, called ellipses, or great curves whose ends never meet, called parabolas and hyperbolas, as shown in Fig. 129; fourth, they do not travel [Pg 114] through space in a line with the planets and Sun, but shoot in and out of our solar system from and to every direction; and fifth and last, they travel at enormously high speeds. ”

“ The hyperbola, Michel, is a curve of the second order, produced by the intersection of a conic surface and a plane parallel to its axis, and constitutes two branches separated one from the other, both tending indefinitely in the two directions. ”

“ The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other. The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. ”

“ If we draw at a point P (fig. 28) on an hyperbola a tangent, the part HK between the asymptotes is bisected at P. The parallelogram PQOQ′ formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ and QP will be the co-ordinates of P, and will satisfy the equation xy = const. = a2. ”

“ Have you any idea how complex a formula must be to trace a curve like this? Just a plain hyperbola is bad enough. This is a test of the machine. Those Mugu boys have worked out this formula to see if she could break it down and draw the equivalent curve, though I don't see how they did it. Even the larger digitals would find this a tough nut to crack, but our baby is a whiz at curves, see? I wonder how they justified the machine-time on it. Of course it is barely possible that they derived the equation themselves, but it must have taken weeks if they did. ”