PARABOLA
, in Geometry, a figure arising from the section of a cone, when cut by a plane parallel to one of its sides, as the section ADE parallel to the side VB of the cone. See Conic Sections, where some general properties are given.
1. From the same point of a cone only one Parabola can be drawn; all the other sections between the Parabola and the parallel side of the cone being ellipses, and all without them hyperbolas. Also the Parabola has but one focus, through which the axis AC passes; all the other diameters being parallel to this, and infinite in length also.
2. The parameter of the axis is a third proportional to any aosciss and its ordinate; viz, AC : CD : : CD : p the parameter. And therefore if x denote any absciss AC, and y the ordinate CD, it will be the parameter; or, by multiplying extremes and means, px = y2, which is the equation of the Parabola.
3. The focus F is the point in the axis where the double ordinate GH is equal to the parameter. Therefore, in the equation of the curve px = y2, taking p = 2y, it becomes 2yx = y2, or 2x = y, that is 2AF = FH, or AF = (1/2)FH, or the focal distance from a vertex AF is equal to half the ordinate there, or = (1/4)p, one-fourth of the parameter.
4. The abscisses of a Parabola are to one another, as the squares of their corresponding ordinates. This is evident from the general equation of the curve px = y2, where, p being constant, x is as y2.
5. The line FE (fig. 2 above) drawn from the focus to any point of the curve, is equal to the sum of the focal distance and the absciss of the ordinate to that point; that is , taking . Or EF is always = EO, drawn parallel to DG, to meet the perpendicular GO, called the Directrix.
6. If a line TBC cut the curve of a Parabola in two points, and the axis produced in T, and BH and CI be ordinates at those two points; then is AT a mean proportional between the abscisses AH and AI, or AT2 = AH . AI .—And if TE touch the curve, then is AT = AD = the mean between AH and AI.
7. If FE be drawn from the focus to the point of contact of the tangent TE, and EK perpendicular to the same tangent; then is FT = FE = FK; and the subnormal DK equal to the constant quantity 2AF or (1/2)p.
8. The diameter EL being parallel to the axis AK, the perpendicular EK, to the curve or tangent at E, bisects the angle LEF. And therefore all rays of light LE, MN, &c, coming parallel to the axis, will be reflected into the point F, which is therefore called the focus, or burning point; for the angle of incidence LEK is = the angle of reflection KEF.
9. If IEK (next fig. below) be any line parallel to the axis, limited by the tangent TC and ordinate CKL to the point of contact; then shall IE : EK : : CK : KL. And the same thing holds true when CL is also in any oblique position.
10. The external parts of the parallels IE, TA, ON, PL, &c, are always proportional to the squares of their intercepted parts of the tangent; that is, the external parts IE, TA, ON, PL, are proportional to CI2, CT2, CO2, CP2, or to the squares CK2, CD2, CM2, CL2.
And as this property is common to every position of the tangent, if the lines IE, TA, ON, &c, be appended to the points I, T, O, &c, of the tangent, and moveable about them, and of such lengths as that their extremities E, A, N, &c, be in the curve of a Parabola in any one position of the tangent; then making the tangent revolve about the point C, the extremities E, A, N, &c, will always form the curve of some Parabola, in every position of the tangent.
The same properties too that have been shewn of the axis, and its abscisses and ordinates, &c, are true of those of any other diameter. All which, besides many other curious properties of the Parabola, may be seen demonstrated in my Treatise on Conic Sections.
In the axis produced take AG = AF (last fig. above) the focal distance, and draw a number of lines EE, EE, &c, perpendicular to the axis AD; then with the distances GD, GD, &c, as radii, and the centre F, describe arcs crossing the parallel ordinates in E, E, &c. Then with a steady hand, or by the side of a slip of bent whale-bone, draw the curve through all the points E, E, E, &c.
If the rule or the directrix BC be laid upon a plane, (first fig. below) with the square GDO, in such manner that one of its sides DG lies along the edge of that rule; and if the thread FMO equal in length to DO, the other side of the square, have one end fixed in the extremity of the rule at O, and the other end in some| point F: Then slide the side of the square DG along the rule BC, and at the same time keep the thread continually tight by means of the pin M, with its part MO close to the side of the square DO; so shall the curve AMX, which the pin describes by this motion, be one part of a Parabola.
And if the square be turned over, and moved on the other side of the fixed point F, the other part of the same Parabola AMZ will be described.
13. If the point of contact C be given: (last fig. above) draw the ordinate CB, and produce the axis till AT be = AB; then join TC, which will be the tangent.
14. Or if the point be given in the axis produced: Take AB = AT, and draw the ordinate BC, which will give C the point of contact; to which draw the line TC as before.
15. If D be any other point, neither in the curve nor in the axis produced, through which the tangent is to pass: Draw DEG perpendicular to the axis, and take DH a mean proportional between DE and DG, and draw HC parallel to the axis, so shall C be the point of contact, through which and the given point D the tangent DCT is to be drawn.
16. When the tangent is to make a given angle with the ordinate at the point of contact: Take the absciss AI equal to half the parameter, or to double the focal distance, and draw the ordinate IE: also draw AH to make with AI the angle HAI equal to the given angle; then draw HC parallel to the axis, and it will cut the curve in C the point of contact, where a line drawn to make the given angle with CB will be the tangent required.
17. To find the Area of a Parabola. Multiply the base EG by the perpendicular height AI, and 2/3 of the product will be the area of the space AEGA; because the Parabolic space is 2/3 of its circumscribing parallelogram.
18. To find the Length of the Curve AC, commencing at the vertex.—Let y = the ordinate BC, p = the parameter, , and ; then shall be the length of the curve AC.
See various other rules for the areas, and lengths of the curve, &c, in my Treatise on Mensuration, sec. 6, pa. 355, &c, 2d edition.
Parabqlas of the Higher Kinds, are algebraic curves, desined by the general equation ; that is, either , or , or , &c.
Some call these by the name of Paraboloids: and in particular, if , they call it a Cubical Paraboloid; if , they call it a Biquadratical Paraboloid, or a Sursolid Paraboloid. In respect of these, the Parabola of the First Kind, above explained, they call the Apollonian, or Quadratic Parabola.
Those curves are also to be referred to Parabolas, that are expressed by the general equation , where the indices of the quantities on each side are equal, as before; and these are called Semi Parabolas: as the Semi Cubical Parabola; or the Semi Biquadratical Parabola; &c.
They are all comprehended under the moregeneral equation , where the two indices on one side are still equal to the index on the other side of the equation; which include both the former kinds of equations, as well as such as these following ones, , or , or , &c.
Cartesian Parabola, is a curve of the 2d order expressed by the equation , containing four infinite legs, viz two hyperbolic ones MM and Bm, to the common asymptote AE, tending contrary ways, and two Parabolic legs MN and DN joining them, being Newton's 66th species of lines of the 3d order, and called by him a Trident. It is made use of by Des Cartes in the 3d book of his Geometry, for finding the roots of equations of 6 dimensions, by means of its intersections with a circle. Its most simple equation is . And points through which it is to pass may be easily found by means of a common Parabola whose absciss is , and an hyperbola whose absciss is d/x; for y will be equal to the sum or difference of the corresponding ordinates of this Parabola and hyperbola.
Des Cartes, in the place abovementioned, shews how to describe this curve by a continued motion. And Mr. Maclaurin does the same thing in a different way, in his Organica Geometria.
Diverging Parabola, is a name given by Newton to a species of five different lines of the 3d order, expressed by the equation .|
The first is a bell-form Parabola, with an oval at its head (fig. 1.); which is the case when the equation , has three real and unequal roots; so that one of the most simple equations of a eurve of this kind is .
The 2d is also a bell-form Parabola, with a conjugate point, or infinitely small oval, at the head (fig. 1.); being the case when the equation has its two less roots equal; the most simple equation of which is .
The third is a Parabola, with two diverging legs, crossing one another like a knot (fig. 2.); which happens when the equation has its two greater roots equal; the more simple equation being .
The fourth a pure bell-form Parabola (fig. 3.); being the case when has two imaginary roots; and its most simple equation is , or .
The fifth a Parabola with two diverging legs, forming at their meeting a cusp or double point (fig. 4); being the case when the equation has three equal roots; so that is the most simple equation of this curve, which indeed is the Semicubical, or Neilian Parabola.
If a solid generated by the rotation of a semi-cubical Parabola, about its axis, be cut by a plane, each of these five Parabolas will be exhibited by its sections. For, when the cutting plane is oblique to the axis, but falls below it, the section is a diverging Parabola, with an oval at its head. When oblique to the axis, but passes through the vertex, the section is a diverging Parabola, having an infinitely small oval at its head. When the cutting is oblique to the axis, falls below it, and at the same time touches the curve surface of the solid, as well as cuts it, the section is a diverging Parabola, with a nodus or knot. When the cutting plane falls above the vertex, either parallel or oblique to the axis, the section is a pure diverging Parabola. And lastly when the cutting plane passes through the axis, the section is the semi-cubical Parabola from which the solid was generated.
PARABOLIC Asymptote, is used for a Parabolic line approaching to a curve, so that they never meet; yet by producing both indefinitely, their distance from each other becomes less than any given line.
There may be as many different kinds of these Asymptotes as there are parabolas of different orders. When a curve has a common parabola for its Asymptote, the ratio of the subtangent to the absciss approaches continually to the ratio of 2 to 1, when the axis of the parabola coincides with the base; but this ratio of the subtangent to the absciss approaches to that of 1 to 2, when the axis is perpendicular to the base. And by observing the limit to which the ratio of the subtangent and absciss approaches, Parabolic Asymptotes of various kinds may be discovered. See Maclaurin's Fluxions, art. 337.
Parabolic Conoid, is a solid generated by the rotation of a parabola about its axis.
This solid is equal to half its circumscribed cylinder; and therefore if the base be multiplied by the height, half the product will be the solid content.
Let BAD be the generating parabola, AC = AT, and BT a tangent at B. Put p = 3.1416, y = BC, x = AC = AT, and then is the curve surface = .
See various other rules and geometrical constructions for the surfaces and solidities of Parabolic Conoids, in my Mensuration, part 3, sect. 6, 2d edition.
Parabolic Pyramidoid, is a solid figure thus named by Dr. Wallis, from its genesis, or formation, which is thus: Let all the squares of the ordinates of a parabola be conceived to be so placed, that the axis shall pass perpendicularly through all their centres; then the aggregate of all these planes will form the Parabolic Pyramidoid.
This figure is equal to half its circumscribed parallclopipedon. And therefore the solid content is found by multiplying the base by the altitude, and taking half the product; or the one of these by half the other.
Parabolic Space, is the space or area included by the curve line and base or double ordinate of the parabola. The area of this space, it has been shewn under the article Parabola, is 2/3 of its circumscribed parallelogram; which is its quadrature, and which was first found out by Archimedes, though some say by Pythagoras.
Parabolic Spindle, is a solid figure conceived to be formed by the rotation of a parabola about its base or double ordinate.
This solid is equal to 8/<*> of its circumscribed cylinder. See my Mensuration, prob. 15, pa. 390, &c, 2d edition.
Parabolic Spiral. See Helicoid Parabola.
Paraboliform Curves, a name sometimes given to the parabolas of the higher orders.
