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sel, &c. are disseminated far and wide. In some plants, as hawk-weed, the pappus adheres immediately to the seed ; in others, as lettuce, it is elevated upon a foot-stalk, which connects it with the seeds. In the first case it is called pappus sessilis; in the second, pappus stipitatus: the foot-stalk, or thread, upon which it is raised, is termed “stipes.” PAR, in commerce, signifies any two things equal in value ; and in money af. fairs, it is so much as a person must give of one kind of specie to render it just equivalent to a certain quantity of another. In the exchange of money with foreign countries, the person to whom a bill is payable is supposed to receive the same value as was paid the drawer by the remitter; but this is not always the case, with respect to the intrinsic value of the coins of different countries, which is owing to the fluctuation in the prices of exchange amongst the several European countries, and the great trading cities. The par, therefore, differs from the course of exchange in this, that the par of exchange shows what other nations should allow in exchange, which is rendered certain and fixed by the intrinsic value of the several species to be exchanged: but the course shows what they will allow in exchange; which is uncertain and contingent, sometimes more, and sometimes less; and hence the exchange is sometimes above, and sometimes under par. See Exch. ANGE. PARABOLA, in geometry, a figure arising from the section of a cone, when cut by a plane parallel to one of its sides. See CoN1c SEcTIons. To describe a parabola in plano, draw a right line A B (Plate Parabola, fig. 1.) and assume a point C without it; then, in the same plane with this line and point, place a square rule D E F, so that the side D E may be applied to the right line A B, and the other E F turned to the side on which the point C is situated. This done, and the thread F G C, exactly of the length of the side of the rule, E F, being fixed at one end to the extremity of the rule F, and at the other to the point C, if you slide the side of the rule, D E, along the right line A B, and by means of a pin, G, continually apply the thread to the side of the rule E F, so as to keep it always stretched as the rule is moved along, the point of this pin will describe the parabola G. H. O. Definitions. 1. The right line A B is called the directrix. 2. The point C is the focus of the parabola. 3. All per
pendiculars to the directrix, as L K, M O, &c. are called diameters; the points, where these cut the parabola are called its vertices; the diameter B I, which passes through the focus C, is called the axis of the parabola; and its vertex, H, the principal vertex, 4. A right line, terminated on each side by the parabola, and bisected by a diameter, is called the ordinate applicate, or simply the ordinate, to that diameter. 5. A line equal to four times the segment of any diameter, intercepted between the directrix and the vertex where it cuts the parabola, is called the latus rectum, or parameter of that diameter, 6. A right line which touches the parabola only in one point, and being produced on each side falls without it, is a tangent to it in that point. Prop. 1. Any right line, as GE, drawn from any point of the parabola, G, perpendicular to A B, is equal to a line G C, drawn from the same point to the focus. This is evident from the description; for the length of the thread, F G C, being equal to the side of the rule E. F., if the part FG, common to both, be taken away, there remains E G = G. C. Q. E. D. . The reverse of this proposition is equally evident, viz. that if the distance of any point from the focus of a parabola be equal to the perpendicular drawn from it to the directrix, then shall that point fall in the curve of the parabola. Prop. 2. If from a point of the parabola, D, (fig. 2.) a right line be drawn to the focus, C; and another, D.A, perpendicular to the directrix; then shall the right line, D E, which bisects the angle A DC, contained between them, be a tangent to the parabola in the point D : a line, also, as H K, drawn through the vertex of the axis, and perpendicular to it, is a tangent to the parabola in that point. . 1. Let any point F, be taken in the line D E, and let FA, FC, and A C be joined; also let F G be drawn perpendicular to the directrix. Then, because (by Prop. 1.) D A = D C, D F common to both, and the angle F D A = F D C, F C will be equal to F A ; but F A greater than F G, therefore F C greater than FG, and consequently the point, F, falls without the parabola : and as the same can be demonstrated of every other point of D E, except D, it follows that D E is a tangent to the parabola in D. Q. E. D. 2. If every point of H K, except H, falls without the parabola, then is H K a tangent in H. To demonstrate this, from any point K, draw KL, perpendicular to A B, and join K C ; then because K C is greater than C II = H B = K, L, it fol. lows that K C is greater than K L, and consequently that the point K falls without the parabola; and as this holds of every other point, except II, it follows that K H is a tangent to the parabola in H. Q. E. D. Prop. 3. Every right line, parallel to a tangent, and terminated on each side by the parabola, is bisected by the diameter passing through the point of contact: that is, it will be an ordinate to that diameter. For let E e (fig. 3 and 4) terminating in the parabola in the points E e, be pool to the tangent I) K; and let A D e a diameter passing through the point of contact D, and meeting E e in L; then shall E L = L e. Let A D meet the directrix in A, and from the points E e, let perpendiculars E F, ef, be drawn to the directrix; let C A be drawn, meeting E e in G and on the centre E, with the distance E C, let a circle be described, meeting A C again in H, and touching the directrix in F; and let D C be joined. Then because D A = DC, and the angle A D'K = the angle CD K, it follows (4.1.) that D K perpendicular to AC; wherefore E e perpendicular to A C, and C G = G H (3.3.); so that e C = e H (4.1.) and a circle described upon the centre e with the radius e C, must pass through H; and because e C = e,f, it must likewise pass through Now because Ff is a tangent to both these circles and A H C cuts them, the square A F = the rectangle C A H (36. 3.) = the square A f, therefore A FAf, and FE, A L, and fe are parallel; and consequently L E = L e. Q. E. D. Prop. 4. If from any point of a parabola, D, (fig. 5.) a perpendicular, DH, be drawn to a diameter B H, so as to be an ordinate to it; then shall the square of the perpendicular, D Ho, be equal to the rectangle contained under the absciss H F, and the parameter of the axis, or to four times the rectangle H F B. 1. When the diameter is the axis; let D H be perpendicular B C, join DC, and draw D A perpendicular A B, and let F be the vertex afthe axis. Then, because H B = D A = D C, it follows that H B
2. When the diameter is not the axis; let E N (fig. 3 and 4) be drawn perpendicular to the diameter A D, and E I, an ordinate to it; and let D be the vertex of the diameter.
Then shall E N* = to the rectangle contained under the absciss, L D, and the parameter of the axis. For let D K be drawn parallel to LE, and consequently a tangent to the parabola in the point D; and let it meet the axis in K : let E F be perpendicular A B the directrix; and on the centre E, with the radius E F, describe a circle which will touch the directrix in F, and pass through the focus C; then join A C, which will meet the circle again in H, and the right lines D K, L E, in the points PG; and, finally, let LE meet the axis in O.
Now since the angles C P K, C B A are right, and the angle B C P common, the triangles C B A, 9 P.K. are equiangular:
Prop. 6. If from any point of a parabola, A, (fig. 6) there be drawn an ordinate, A C, to the diameter B C ; and a tangent to the parabola in A, meeting the diameter in D : then shall the segment of the diameter, CD, intercepted between the ordinate and the tangent, be bisected in the vertex of the diameter B. For let B E be drawn parallel to A D, it will be an ordinate to the diameter A. E.; and the obsciss B C will be equal to the absciss A E, or B D. Q. E. 1). Hence, if A C be an ordinate to B C, and A D be drawn so as to make B D = DC, then is A D a tangent to the parabo|a. Also the segment of the tangent, A D, intercepted between the diameter and Point of contact, is bisected by a tangent BG, passing through the vertex of i) C. “To draw Tangents to the Parabola.” If the point of contact C be given (fig. 7.): draw the ordinate C B, and produce the axis till A T be = A B ; then join TC, which will be the tangent, Or if the Point be given in the axis produced: take B = A T, and draw the ordinate B C, which will give C the point of contact; to which draw the line t c as before. If D * any other point, neither in the curve or in the axis produced, through which the tangent is to pass, draw D'E G perPendicular to the axis, and take 1) H *, nean proportional between D E and DG, and draw H c parallel to the axis, *śhall C be the point of contact through Yooh, and the given point D, the tangent C T is to be drawn.
When the tangent is to make a given *gle with the ordinate at the point of Sontact: take the absciss A I equal to half the Parameter, or to double the focal dis*e, and draw the ordinate I E: also o H to make with A. I the angle * I equal to the given angle; then
*W. H. C. parallel to the axis, and it will * the curve in C, the point of contact, *re a line drawn to make the given
p X (7 s -- hyp. log of q + s) be the length of the curve A C. PARAbol A, Cartesian, is a curve of the second order, expressed by the equation a: y = a a 3 + b ar” + c a' + d, containing four infinite legs, viz. two hyperbolic ones M. M., B m (Plate Parabola, fig. 8), (A E being the asymptote) tending contrary ways, and two parabolic legs R N, M N joining them, being the sixty-sixth species of lines of the third order, according to Sir Isaac Newton, called by him a trident : it is made use of by Des Cartes, in the third book of his Geometry, for finding the roots of equations of six dimensions by its intersections with a circle. Its most simple equation is a y = x3 + as, and the points through which it is to pass may be easily found by means of a common parabola, whose absciss is a Jo + b x + c,
for y will be equal to the sum or difference of the correspondent ordinates of this parabola and hyperbola. PARAbol A, diverging, a name given by Sir Isaac Newton to five different lines of the third order, expressed by the equation y y = a +3 + b x* + c < -H, d. PARABOLIC asymptote, in geometry, 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. Maclaurin observes, that 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 two to one, when the axis of the parabola coincides with the base; but this ratio of the subtangent to the absciss approaches to that of one to two, 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. PARAbolic conoid, in geometry, a solid generated by the rotation of a parabola about its axis: its solidity is - 3 of that of its circumscribing cylinder. The circles, conceived to be the elements of this figure, are in arithmetical proportion, decreasing towards the vertex. A parabolic conoid is to a cylinder of the same base and height, as 1 to 2, and to a cone of the same base and height as 1} to 1. See the article GAUGING. PAn Anolic cuneus, a solid figure formed by multiplying all the DB's (Plate Parabola, fig. 9.) into the D S's; or, which amounts to the same, on the base A PB erect a prism whose altitude is A S ; this will be a parabolical cuneus, which of necessity will be equal to the parabolical pyramidoid, as the component rectangles in one are severally equal to all the component squares in the other. PARAbolic pyramidoid, a solid figure, generated by supposing all the squares of the ordinate applicates in the parabola so placed, as that the axis shall pass through all the centres at right angles; in which case, the aggregate of the planes will form the parabolical pyramidoid. The solidity hereof is had by multiplying, the base by half the altitude, the reason of which is obvious; for the compoment planes being a series of arithmetical proportionals beginning from 0, their sum will be equal to the extremes multiplied by half the number of terms. PARAholic space, the area contained between any entire ordinate, as V V (Plate Parabola, fig. 10.), and the curve of the incumbent parabola. The parabolic space is to the rectangle of the semi-ordinate into the absciss, as 2 to 3; to a triangle inscribed on the ordimate as a base, it is as 4 to 3. Every parabolical and paraboloidical space is to the rectangle of the semi-ordinate into the absciss, as r a y (m + r.) to a y ; that is, as r to m + r. PARAbolic spindle, in gauging ; a cask of the seoond variety is called the middle frustrum of a parabolic spindle. The parabolic spindle is eight-fifteenths of its circumscribing cylinder. PARADE, in war, is a place where the troops meet to go upon guard, or any other service. In a garrison, where there are two, three, or more regiments, each have their parade appointed, where they are to meet upon all occasions, especially upon any alarm. And in a camp, all parties, convoys, and detachments, have a
o place appointed them at the ead of some regiment. PARADISEA, the bird of Paradise, in natural history, a genus of birds of the order Picas. Generic character: bill covered at the base with downy feathers; nostrils covered by the feathers: tail of ten feathers, two of them, in some species, very long ; legs and feet very large and strong. These birds chiefly inhabit North Guinea, from which they migrate in the dry season into the neighbouring islands. They are used in these countries as ornaments for the head-dress, and the Japanese, Chinese, and Persians, import them for the same purpose. The rich and great among the latter attach these brilliant collections of plumage, not only to their own turbans, but to the housings and harnesses of their horses. They are found only within a few degrees of the equator. Gmelin enume. rates twelve species, and Latham eight. P. apoda, or the greater Paradise bird, is about as large as a thrush. These birds are supposed to bread in North Guinea, whence they migrate into Aroo, return. ing to North Guinea with the wet monsoon. They pass in flights of thirty or forty, headed by one whose flight is higher than that of the rest. They are often distressed by means of their long feathers in sudden shiftings of the wind, and, unio ble to proceed in their flight, are easily taken by the natives, who also catch them with birdlime, and shoot them with blunt: ed arrows. They are sold at Aroo for an iron nail each, and at Banda for half a ro dollar. Their food is not ascertained, and they cannot be kept alive in confino. ment. The smaller bird of Paradise is supposed by Latham to be a mer." riety of the above. It is found only. the Papuan islands, where it is caught by the natives often by the hand, * exenterated and seared with a hot * in the inside, and then put into the hol. low of a bamboo to secure its plum"#" from injury. PARADOX, in philosophy, a prop". tion seemingly absurd, as being com". to some received opinion; but yet ". in fact. No science abounds more." paradoxes than geometry; thus, "... right line should continually app". to the hyperbola, and yet never o it, is a true paradox; and in the so manner, a spiral may continually o proach to a point, and yet not e^* ! in any number of revolutions, how" reat.
whereby a letter or syllable is added to the end of a word : as med, for me, dicier, for dici, &c. PARALLACTIC, in general, something relating to the parallax of heavenly bodies. See PARALLAx. The parallactic angle, of a star, &c. is the difference of the angles C E A (Plate Parabola, &c. fig. 11.) B T A, under which its true and apparent distance from the zenith is seen; or, which is the same thing, it is the angle T S E. The sines of the parallactic angle A L T, A S T, (fig. 12) at the same or equal distances, ZS, from the zenith, are in the reciprocal ratio of the distances T. L., and T S, from the centre of the earth. PARALLAX, in astronomy, denotes a change of the apparent place of any heaWenly body, caused by being seen from different points of view; or it is the dif. ference between the true and apparent distance of any heavenly body from the zenith. Thus let A B (Plate XII. Miscell. fig. 1.) be a quadrant of great circle on the earth's surface, A, the place of the spectator, and the point V, in the heavens, the vertex and zenith. Let V N H represent the starry firmament, AD the sensible horizon in which suppose the star C to be seen, whose distance from the centre of the earth is T C. \f this star were observed from the centre T, it would appear in the firmament in E, and elevated above the hori20m by the arch D E ; this point E is called the true place of the phenomenon or star. But an observer viewing it from the surface of the earth at A, will see it at D, which is called its visible or appasent place; and the arch DE, the distance between the true and visible place, is whatastronomers call the parallax of the Star, or other phenomenon. If the star rise higher above the horizon to M, its true place visible from the Centre is P, and its apparent place N, whence its parallax will be the arch PN, which is less than the arch D E. The horizontal parallax, therefore, is the greatest; and the higher a star rises, the less is its parallax; and if it should come to the vertex or zenith, it would have no Parallax at all; for when it is in Q, it is seen both from T and T and A in the same line T A V, and there is no difference between its true and apparent or visible place. Again, the further a star is distant from the earth, so much the less ls its parallax ; thus the parallax of the WOL. IX.
star F is only G D, which is less than. D E, the parallax of C. Hence it is plain, that the parallax is the difference of the distances of a star from the zenith, when seen from the centre and from the sur. face of the earth ; for the true distance of the star M from the zenith is the arch V P, and its apparent distance V N, the difference between which, P N, is the parallax. These distances are measured by the angles W T M, and V A M, but V A MV T M = T M A. For the external angle V A M = angle A T M + angle A M T, the two inward and opposite angles; so that A M T measures the parailax, and upon that account is itself . frequently called the parallax ; and this is always the angle under which the semidiameter of the earth, A T, appears to an eye placed in the star; and therefore, where the semi-diameter is seen directly, there the parallax is greatest, viz. in the horizon. When the star rises higher, the sine of the parallax is always to the sine of the star's distance from the zenith, as the semi-diameter of the earth to the distance of the star from the earth’s centre; hence if the parallax of a star be known at any one distance from the zenith, we can find its parallax at any other distance. If we have the distance of a star from the earth, we can easily find its parallax ; for on the triangle T A C, rectangular at A, having the semi-diameter of the earth, and T C the distance of the star, the angle A C T, which is the horizontal parallax, is found by trigonometry; and on the other hand, if we have this parallax, we can find the distance of the star; since in the same triangle, having A T, and the angle A CT, the distance T C may be easily found. Astronomers, therefore, have invented several methods for finding the parallaxes of stars, in order thereby to discover their distances from the earth. However, the fixed stars are so remote as to have no sensible parallax; and even the sun, and all the primary planets, except Mars and Venus when in perigee, are at so great distances from the earth, that their parallax is too small to be observed. In the moon, indeed, the parallax is found to be very considerable, which in the horizon amounts to a degree or more, and may be found thus: in an eclipse of the moon, observe when both its horns are in the same vertical circle, and at that inQ -