Parabola

“ That the stone moves exactly in a parabola is a result of the given conditions and follows necessarily from them. The form of the parabola belongs to the whole phenomenon as much as any other feature of it. The hypothetical mind described above which has no need of the roundabout way of thought, would find itself presented, not only with a sequence of visual percepts at different points, but, as part and parcel of these phenomena, also with the parabolic form of the line of flight, which we can add to the phenomenon only by an act of thought. ”

“ Parabola, par-ab′o-la, n. (geom.) a curve or conic section, formed by cutting a cone with a plane parallel to its slope (for illustration, see Cone) .—adjs. Parabol′ic; Parabol′iform.—n. Parab′oloid, the solid which would be generated by the rotation of a parabola about its principal axis. [Gr. parabolē; cf. Parable.]
Parabolanus, par-a-bō-lā′nus, n. in the early Eastern Church, a lay assistant to the clergy for waiting on the sick. [Gr. parabolos, reckless.] ”

“ There was only one parabola, and that was formed when the side of the cone was parallel to the plane of section, that is when the falling grease just touched the edge of the candlestick; there is only one bubble with no pressure, the catenoid, and this is drawn by rolling the parabola. As the cone is gradually inclined more, so the sections become at first long ellipses, which gradually become more and more round until a circle is reached, after which they become more and more narrow until a line is reached. ”

“ For the axis of the parabola along which we imagine the natural motion of a falling body to take place stands perpendicular to a horizontal surface and ends at the center of the earth; and since the parabola deviates more and more from its axis no projectile can ever reach the center of the earth or, if it does, as seems necessary, then the path of the projectile must transform itself into some other curve very different from the parabola. ”

“ But the true answer rather is that in conceiving as such we do not necessarily imagine a man or a tree at all, any more than—if such an illustration may serve—in writing the equation to the parabola we necessarily draw a parabola as well. ”

“ 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. ”

“ 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. ”

“ The center of the parabola is at infinity, and therefore all its diameters are parallel, for the pole of a tangent line is the point of contact.
The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords. ”

“ In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpendicular let fall from the focus on the tangent.
Cor. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the distance. ”

“ All that has been said above concerning the amount of momentum in the projectile at the extremity of the parabola must be understood to refer to a blow received on a line at right angles to this parabola or along the tangent to the parabola at the given point; for, even though the motion has two components, one horizontal, the other vertical, neither will the momentum along the horizontal nor that upon a plane perpendicular to the horizontal be a maximum, since each of these will be received obliquely. ”

“ In the article Geometry, Analytical, it is shown that the general equation of the second degree represents a parabola Analytic Geometry. when the highest terms form a perfect square. This is the analytical expression of the projective property that the line at infinity is a tangent. The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form where 2a=semilatus rectum ”

“ The Parabola is the curve of the section made by a plane passing through a cone parallel to one of its sides.
The Hyperbola is the curve of a section made by a plane passing through a cone parallel to its axis, or inclined at a greater angle to its base than its side, but not through its apex.
Fig. 31 The elevation of a cone is shown at A B C. A section through point X at right angles to the axis of the cone is a Circle. A section passing through and across the cone from point X, but not at right angles to the axis, is an Ellipse, as at X 1. ”

“ A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola. Sagr.
Here, Salviati, it will be necessary to stop a little while for my sake and, I believe, also for the benefit of Simplicio; for it so happens that I have not gone very far in my study of Apollonius and am merely aware of the fact that he treats of the parabola and other conic sections, without an understanding of which I hardly think one will be able to follow the proof of other propositions depending upon them. ”

“ But a body which falls from rest at b along the line bd will during the same time-interval fall through the height of the parabola [285] bd. Hence a body falling from rest at a, turned into a horizontal direction with the speed ab will traverse a space equal to dc. Now if one superposes upon this motion a fall along bd, traversing the height bd while the parabola bc is described, then the momentum of the body at the terminal point c is the resultant of a uniform horizontal momentum, whose value is represented by ab, and of another momentum acquired by fall from b to the terminal point d or c ”

“ 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. ”

“ This would not be the occasion for any allusion to the geometrical properties of this curve; they should be sought in works on mathematics. It will here be only necessary to point to the connection which exists between the parabola and the ellipse. In a former chapter we have explained the construction of the latter curve, and we have shown how it possesses two foci. Let us suppose that a series of ellipses are drawn, each of which has a greater distance between its foci than the preceding one. ”

“ There is no finality in human history. It is folly to believe that any religions, any social orders, any scientific hypotheses, are more than provisional, and partially possessed of truth. Let us assume that the whole curve of human existence on this planet describes a parabola of some twenty millions of years in duration. ”

“ 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. ”

“ Bodies moving under the influence of {35} gravitation can move in any of these curves, but only the circle and ellipse are closed orbits. A particle moving in a parabola or hyperbola can only make one approach to its attracting body; after such approach it continually recedes from it. As the circle and parabola are only the two extreme forms of an ellipse, the two foci being at the same point for the circle and at an infinite distance apart for the parabola, we may regard all orbits under gravitation as being ellipses of one form or another. ”

“ 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. ”

“ Except under certain limited conditions to be determined later a curve with infinite ordinates can not be of great statistical value.
The parabola, κ2-3λ = 0, obtained by equating the discriminant of this quadratic to zero separates the points on the (λ, κ) plane which correspond to curves of no infinite points from those corresponding to curves of two infinite points.
Types of Curves
Therefore, all pairs of values of λ and κ within the parabola, with the exception of the very narrow region also within the first discriminant curve, give uni-modal curves without infinite ordinates. ”

“ It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves ”

“ To produce such an effect, a body ought to be accelerated by an impelling medium, and not impeded by a resisting one.
Example 2. Let the line PFQ be a parabola, having its axis AF perpendicular to the horizon PQ, to find the density of the medium, which will make a projectile move in that line.
From the nature of the parabola, the rectangle PDQ is equal to the rectangle under the ordinate DI and some given right line; that is, if that right line be called b; PC, a; PQ, c ”

“ 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. ”

“ A man may form to himself the notion of a parabola, or a cycloid, from the mathematical definition of those figures, although he had never seen them drawn or delineated. Another, who knows nothing of the mathematical definition of the figures, may see them delineated on paper, or feel them cut out in wood. Each may have a distinct conception of the figures, one by mathematical reasoning, the other by sense. ”

“ If I tilt up the plane and make the force of gravity big, the parabola is long and thin;88 if I weaken down the force of gravity by making the plane nearly horizontal, then it is wide and flat.
One can also show this by a stream of peas or shot. The little bullets go each with a uniform velocity horizontally, and an accelerated force downwards.
Instead of peas we can use water. A stream of it rushing horizontally out of an orifice will soon bend down into a parabola. ”

“ Chap. 6. The Author considering; that Vincenzo Viviani in Book De Maximis & Minimis found, that if there were innumerable Parabolas described, having the same Axis and Vertex common, if from any point in that Axis, the shortest Lines were drawn to those Parabola's, all those points of Incidence would fall in an Ellipsis; and the Authors Analysis taught him, that the Prop. was Universal, wheresoever the point be assigned, from which the least lines are to be drawn; which he hath extended, and applyed to those infinite sorts of other Parabola's. ”

“ To produce a true and uniform curve is, however, the acme of troubles, whether it is desired to obtain the spherical, or parabola, or any other curve. It is generally supposed that the spherical curve is a very easy matter, so easy indeed that it is difficult to avoid. This is a very great mistake—a spherical curve of undeviating truth is as difficult a problem as a true parabola. ”

“ For it may be observed, that the curvilinear paths ascribed to military projectiles by Rivius and Tartalea, and by other writers who followed them, as Digges and Norton in our own country, though utterly different from the theoretical form, the parabola, do, in fact, approach nearer the true paths of a cannon or musket ball than a parabola would do; and this approximation more especially exists in that which at first sight appears most absurd in the old theory; namely, the assertion that the ball, which ascends in a sloping direction, finally descends vertically. ”