# Parabola Sentence Examples

By joining the points so obtained the

**parabola**may be described.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).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.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.In a vacuum, the projectile acted on by the force of projection begins to fall under the action of gravity immediately it leaves the bore, and under the combined action of these two forces the path of the projectile is a

**parabola**.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.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.The curve of the main arch is a

**parabola**.Now, except for very short bridges and very unequal loads, a

**parabola**can be found which includes the curve of maximum moments.AdvertisementExperience shows that (a) a

**parabola**having the same ordinate at the centre of the span, or (b) a**parabola**having 15 ons FIG.This curve is a

**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.AdvertisementThe 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.Then if a pencil be placed along B C so as to keep the string taut, and the limb AB be slid along the directrix, the A pencil will trace out the

**parabola**.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.AdvertisementIn 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.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 y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition.Expressing this condition we obtain mb = 1/ nc = o as the relation which must hold between the co-efficients of the above equation and the sides of the triangle of reference for the equation to represent a

**parabola**.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.The cartesian

**parabola**is a cubic curve which is also known as the trident of Newton on account of its three-pronged form.AdvertisementThis is sometimes termed the campaniform (or bell-shaped)

**parabola**.It may be remarked that since this line joins homologous points of two similar rows it will envelope a

**parabola**.The curve of positions corresponding to (4) is a

**parabola**, and that of velocities is a straight line.This is a

**parabola**with vertical axis, of latus-rectum 2uiulg.The case n =2 gives the

**parabola**as before.The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an ellipse,

**parabola**or hyperbola.Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a

**parabola**with an acceleration g sin a/(I+C/Ma)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.The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and

**parabola**respectively.The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the

**parabola**as projections of the ellipse.The two legs of a parabolic branch may converge to ultimate parallelism, as in the conic

**parabola**, or diverge to ultimate parallelism, as in the semi-cubical**parabola**y 2 = x 3, and the branch is said to be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical**parabola**y = x 3 .As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the

**parabola**they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity.A body moving in a

**parabola**or hyperbola would recede indefinitely from its centre of motion and never return to it.The cissoid is the first positive pedal of the

**parabola**y2+8ax=o for the vertex, and the inverse of the**parabola**y 2 = 8ax, the vertex being the centre of inversion, and the semi-latus rectum the constant of inversion.This is the equation to a

**parabola**, and is equivalent to the empirical formula of Avenarius, with this difference, that in Tait's formula the constants have all a simple and direct interpretation in relation to the theory.When the question is tested more carefully, either by taking more accurate measurements of temperature, or by extending the observations over a wider range, it is found that there are systematic deviations from the

**parabola**in the majority of cases, which cannot be explained by errors of experiment.In ancient geometry the name was restricted to the three particular forms now designated the ellipse,

**parabola**and hyperbola, and this sense is still retained in general works.One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse,

**parabola**and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a

**parabola**; and if less than unity, an ellipse.A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a

**parabola**, and if within the circle a hyperbola.The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and

**Parabola**.Archimedes contributed to the knowledge of these curves by determining the area of the

**parabola**, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces.When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the

**parabola**; if the inclination of the plane be greater than that of the side, we obtain the hyperbola.Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the

**parabola**, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.The word

**parabola**was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a

**parabola**.The focus of the

**parabola**was discovered by Pappus, who also introduced the notion of the directrix.There was a dead silence of suspense among the crowd as the ball described a lofty

**parabola**.In 1657 he became the first to find the arc length of an algebraic curve when he rectified the cubical

**parabola**.The first (a mechanical proof) begins, after some preliminary propositions on the

**parabola**, in Prop. 6, ending with an integration in Prop. 16.A Latin version of them was published by Isaac Barrow in 1675 (London, 4to); Nicolas Tartaglia published in Latin the treatises on Centres of Gravity, on the Quadrature of the

**Parabola**, on the Measurement of the Circle, and on Floating Bodies, i.If the law of attraction is that of gravitation, the orbit is a conic section - ellipse,

**parabola**or hyperbola - having the centre of attraction in one of its foci; and the motion takes place in accordance with Kepler's laws (see Astronomy).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.Then (see

**Parabola**) Tc = Cv, Av = Vb, and K ab is parallel to AB, so that aC = Cb.At any point x from the abutment, the bending moment is M = Zwx(l - x), an equation to a

**parabola**.In the ancient theory due to Galileo, the resistance of the air is ignored, and, as shown in the article on Mechanics (§ 13), the trajectory is now a

**parabola**.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.The

**parabola**is the curve described by a projectile which moves in a non-resisting medium under the influence of gravity (see Mechanics).To construct the

**parabola**when the focus and directrix are given, draw the axis CD and bisect CF at G, which gives the vertex.The orthocentre of a triangle circumscribing a

**parabola**is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix ("Steiner's Theorem").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).The envelope of this last equation is 27ay 2 =4(x-2a) 3, which shows that the evolute of a

**parabola**is a semi-cubical**parabola**(see below Higher Orders).The equation to a

**parabola**in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, the equation to the general conic circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a**parabola**the line as + b/ + cy = o must be a tangent.In the fourth of Galileo's dialogues on mechanics, he demonstrated that the path described by a projectile, being the result of the combination of a uniform transverse motion with a uniformly accelerated vertical motion, must, apart from the resistance of the air, be a

**parabola**.Menaechmus discussed three species of cones (distinguished by the magnitude of the vertical angle as obtuse-angled, right-angled and acuteangled), and the only section he treated was that made by a plane perpendicular to a generator of the cone; according to the species of the cone, he obtained the curves now known as the hyperbola,

**parabola**and ellipse.In his extant Conoids and Spheroids he defines a conoid to be the solid formed by the revolution of the

**parabola**and hyperbola about its axis, and a spheroid to be formed similarly from the ellipse; these solids he discussed with great acumen, and effected their cubature by his famous "method of exhaustions."Kepler's greatest contribution to geometry lies in his formulation of the "principle of continuity" which enabled him to show that a

**parabola**has a "caecus (or blind) focus" at infinity, and that all lines through this focus are parallel (see Geometrical Continuity).In the Quadrature of the

**parabola**Archimedes finds the area of a segment of a**parabola**cut off by any chord.While resembling the

**parabola**in extending to infinity, the curve has closest affinities to the ellipse.The line TCV is parallel to the axis of the

**parabola**.Similarly, for a corresponding figure K'L'BA outside the

**parabola**, the area is lK'L'(K'A+4M'C +L'B).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.This

**parabola**is the curve of maximum moments for a travelling load uniform per ft.If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a

**parabola**curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid.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 graph of F is a straight line; that of M is a

**parabola**with vertical axis.This property is characteristic of a

**parabola**whose axis is vertical.The cartesian equation to a

**parabola**which touches the coordinate axes is 1 / ax+'1 / by= i, and the polar equation when the focus is the pole and the axis the initial line is r cos 2 6/2 = a.