It was investigated by Galileo, who erroneously determined it to be a parabola; Jungius detected Galileo's error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Aeta Eruditorum.
- The parabola affords a simple example of the use of infinitesimals.
6) be any arc of a parabola; and suppose we require the area of the figure bounded by this arc and the chord AB.
Thus we find from (i) that Simpson's second formula, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us).
By joining the points so obtained the parabola may be described.
- Geometrical constructions of the parabola are to be found in T.
Similarly, the inertia parallel to Oy and Oz is NW' - 1 B W', B C (b2 +-X, c 2 ab and A +C abc/ZP, Ao For a sphere a=b=c, Ao= Bo=Co =, 'a' = Q = = z, (9) U from (II), (16) so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration W - W', g (30) W+ aW Thus a spherical air bubble, in which W/W' is insensible, will begin to rise in water with acceleration 2g.
It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference.
Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and M draw the tangent at C, meeting AT and BT in a and b.
Similarly, for a corresponding figure K'L'BA outside the parabola, the area is lK'L'(K'A+4M'C +L'B).
- In § 23 the area of a right trapezium has been expressed in terms of the base and the two sides; and in § 34 the area of a somewhat similar figure, the top having been replaced by an arc of a parabola, has been expressed in terms of its base and of three lengths which may be regarded as the sides of two separate figures of which it is composed.
The top is then a parabola whose axis is at right angles to the base; and the area can therefore (§ 34) be expressed in terms of the two bounding ordinates and the midordinate.
Hence, for the case of a parabola, we can express the area in terms of the bounding ordinates of two strips, but, if we use mid-ordinates, we require three strips; so that, in each case, three ordinates are required.
These problems were also attacked by the Arabian mathematicians; Tobit ben Korra (836-901) is credited with a solution, while Abul Gud solved it by means of a parabola and an equilateral hyperbola.
A solution by means of the parabola and hyperbola was given by Dionysodorus of Amisus (c. 1st century B.c), and a similar problem - to construct a segment equal in volume to a given segment, and in surface to another segment - was solved by the Arabian mathematician and astronomer, Al Kuhi.
As the load travels, the shear at the head of the train will be given by the ordinates of a parabola having its vertex at A, and a maximum F max.
Now, except for very short bridges and very unequal loads, a parabola can be found which includes the curve of maximum moments.
Experience shows that (a) a parabola having the same ordinate at the centre of the span, or (b) a parabola having 15 ons FIG.
3 parallel to AM; the energycurve AQE would be another straight line through A; the velocitycurve AvV, of which the ordinate v is as the square root of the energy, would be a parabola; and the acceleration of the shot being constant, the time-curve AtT will also be a similar parabola.
But if the pressure-curve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energy-curve will be a parabola curving upwards, and the velocity-curve a hyperbola with center at F'.
He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.
It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names parabola, ellipse, hyperbola.
He went on to deal with the case of projectiles, and was led to the conclusion that the motion in this case could be regarded as the result of superposing a horizontal motion with uniform velocity and a vertical motion with constant acceleration, the latter identical with that of a merely falling body; the inference being that the path of a projectile would be a parabola except for deviations attributed to contact with the air, and that in a vacuum this path would be accurately followed.
In this case the curve representing the distribution of temperature is a parabola, and the conductivity k is deduced from the mean rise of temperature (R-R°)/aR° by observing the increase of resistance R-R° of the bar, and the current C. It is also necessary to measure the cross-section g, the length 1, and the temperature-coefficient a for the range of the experiment.
(9) By a property of the parabola, the mean temperature is 3rds of the maximum temperature, we have therefore (R - R 0)/aRo =1C 2 Ra/12gk, (io) which gives the conductivity directly in terms of the quantities actually observed.
PARABOLA, a plane curve of the second degree.
The general relations between the parabola, ellipse and hyperbola are treated in the articles Geometry, Analytical, and Conic Sections; and various projective properties are demonstrated in the article Geometry, Projective.
Here only the specific properties of the parabola will be given.
Any number of points on the parabola are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal.
It also follows that a line half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the ninepoint circle of a self-conjugate triangle passes through the focus.
In the article Geometry, Analytical, it iS Shown that the general equation of the second degree represents a parabola when the highest terms form a perfect square.
This property is characteristic of a parabola whose axis is vertical.
Which is the equation of the parabola in question.
The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.
And at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical.
Parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D.
The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ...
The graph of F is a straight line; that of M is a parabola with vertical axis.
More convenient forms in terms of a single parameter are deduced by substituting x' =am t, y' = aam (for on eliminating in between these relations the equation to the parabola is obtained).
Try = o to be a parabola is lbc+mca+nab = o, and the conic for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o.
In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.
The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form.