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 affairs, 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 EXCHANGE.

PARABOLA, in geometry, a figure arising from the section of a cone, when cut by a plane parallel to one of its sides. See CONIC 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, É 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 G E, 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 F G, 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, DE, which bisects the angle A D C, contained between them, be a tangent to the parabola in the point D: a line, also, as HK, 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 DE, and let FA, FC, and AC be joined; also let F G be drawn perpendicular to the directrix. Then, because (by Prop. 1.) DA = DC, 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 F G, 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 HK a tangent in H. To demonstrate this, from any point K, draw K L, perpendicular to A B, and join K C; then because K C is

greater than CH=HB = KL, it follows 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 H, 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 Ee (fig. 3 and 4) terminating in the parabola in the points Ee, be parallel to the tangent D K; and let A D be a diameter passing through the point of contact D, and meeting E e in L; then shall E L = Le.

Let A D meet the directrix in A, and from the points Ee, let perpendiculars EF, ef, be drawn to the directrix; let C A be drawn, meeting Ee 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= D C, and the angle A D K = the angle CDK, 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 eCe H (4. 1.) and a circle described upon the centre e with the radius e C, must pass through H; and because eCef, it must likewise pass through f. Now because Ff is a tangent to both

=

these circles and A H C cuts them, the square AF the rectangle CAH (36. 3.) the square Af; therefore A F= Af, and FE, A L, and fe are parallel; and consequently LE Le. Q. E. D. Prop. 4. If from any point of a parabola, D, (fig. 5.) a perpendicular, D H, be drawn to a diameter B H, so as to be an ordinate to it; then shall the square of the perpendicular, DH, 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 DH be perpendicular B C, join D C, and draw D A perpendicular A B, and let F be the vertex af the axis. Then, because HB = DA=D C, it follows that H B = DC'D H+H C. Likewise, because B FF C, H B 4 times the rectangle H FC+HC (by 8. 2). Wherefore DH+H C' = 4 times the rectangle H FB+HC; and D H1 = 4 times the rectangle H F B; that is, D H' the rectangle contained under the absciss H F, and the parameter of the

axis.

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 L 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 L E, 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, LE, in the points PG; and, finally, let LE meet the axis in 0.

Now since the angles C P K, CBA are right, and the angle B C P common, the triangles C B A, C P K are equiangular; and AC CB (or CK: CP) :: OK: GP; and ACXG P=0KXC B. Again, because C A 2 C P, and CH 2C G, AH 2 GP; and consequently the rectangle CA H=CA×2 GP = OK × 2 C B. But, E N = F A2 = rectangle CA H; and consequently, EN OK X 2 C B the rectangle contained under the absciss L D, and the parameter of the axis. Q. E. D.

=

Hence, 1. The squares of the perpendiculars, drawn from any points of the parabola to any diameters, are to one anothe as the abscissæ intercepted between the vertices of the diameters and the ordinates applied to them from the same points.

2, The squares of the ordinates, applied to the same diameter, are to each other as the abscissæ between each of them and the vertex of the diameter. For let E L, QR be ordinates to the same diameter DN; and let EN, QS be perpendiculars to it. Then, on account of the equiangular triangles EL N, QRS, E L3 : QREN: QS: that is, as the ab. sciss D L to the absciss D R.

Prop. 5. If from any point of a parabola E (fig. 3 and 4), an ordinate E L, be applied to the diameter A D; then shall the square EL be equal to the rectangle contained under the absciss D L, and the latus rectum or parameter of that di

ameter.

For since QR = DK, Q R will be equal to D M + M K2; but (by case 1. of Prop. 4), D M2 = 4 times the rectangle MQ B: and because M Q = : Q K, MK 4 M Q': wherefore Q R = 4

times the rectangle MQB + 4 MQ; that is, to 4 times the rectangle Q M B. But M Q = Q K = DR, and MB = DA; wherefore Q R 4 times the rectangle RDA and because Q R, E L are ordinates to the diameter A D, Q R (by cor. 2, of Prop. 4), E L' (: RD: LD) :: 4 times the rectangle R DA: 4 times

:

angle with CB will be the tangent required.

"To find the Area of a Parabola." Multiply the base E G by the perpendicular height A I, and of the product will be the area of the space A E GA; because the parabolic space is of its circumscrib

=

4 times the rectangle L D A, or the rectangle contained under the absciss L D, and the parameter of the diameter A D: and from this property Apollonius called the curve a parabola. Q. E. D.

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, C D, intercepted between the ordinate and the tangent, be bisected in the vertex of the diameter B. For let BE be drawn parallel to A D, it will be an ordinate to the diameter A E; and the absciss B C will be equal to the absciss A E, or B D. Q. E. D.

Hence, if A C be an ordinate to BC, and A D be drawn so as to make B D

=

DC, then is A D a tangent to the parabola. Also the segment of the tangent, AD, intercepted between the diameter and point of contact, is bisected by a tangent BG, passing through the vertex of Ï 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 T C, which will be the tangent, Or if the point be given in the axis produced: take ABAT, 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 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 D G, and draw H C parallel to the axis, so shall C be the point of contact through which, and the given point D, the tangent DC T is to be drawn.

When the tangent is to make a given angle with the ordinate at the point of contact: take the absciss A I equal to half the parameter, or to double the focal distance, and draw the ordinate I E: also draw A H to make with AI the angle HAI equal to the given angle; then draw H C parallel to the axis, and it will cut the curve in C, the point of contact, where a line drawn to make the given

To find the length of the Curve A C," commencing at the vertex. Let y the ordinate B C, p the parameter,

=

2y 9= and s = √1 × q'; then shall

8

& px (q + hyp. log. of q+s) be the length of the curve & C.

PARABOLA, Cartesian, is a curve of the second order, expressed by the equation yax3+ bx + cx+d, containing four infinite legs, viz. two hyperbolic ones M M, B m (Plate Parabola, fig. 8.), (AE being the asymptote) tending contrary ways, and two parabolic legs B 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 findby its intersections with a circle. Its most ing the roots of equations of six dimensions simple equation is x y=x3+ a3, and the points through which it is to pass may be easily found by means of a common parabola, whose absciss is a 22 + bx + c,

d

and an hyperbola, whose absciss is -; for y will be equal to the sum or difference of the correspondent ordinates of this parabola and hyperbola,

PARABOLA, diverging, a name given by Sir Isaac Newton to five different lines of the third order, expressed by the equation y ya x3 + b x2 + c x + 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 dis

tance 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

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 TS E. The sines of the parallactic angle A LT, AS T, (fig. 12.) at the same or equal distances, Z S, from the zenith, are in the recip. rocal 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 heavenly 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 NH 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. If this star were observed from the centre T, it would appear in the firmament in E, and elevated above the horizon 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 apparent place; and the arch D E, the distance between the true and visible place, is what astronomers 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 P N, 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 is its parallax; thus the parallax of the YOL. IX.

star F is only G D, which is less than DE, 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 surface 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 V T M, and V A M, but V A M-VTM TMA. For the external angle VA M = angle A T M + angle A M T, the two inward and opposite angles; so that A M T measures the parallax, 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 AC 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 in૨