The British Encyclopedia, Or Dictionary of Arts and Sciences

la a toleration, under certain regulations, of the religious worship of the Catholics, qualifying in like manner, and of their schools for education; the fourth enacts, that no one shall be summoned to take the oath of supremacy prescribed by statutes 1 William and Mary, st. 1. c. 8.; 1George I. st. 2. c. 15; or the declaration against transubstantiation required by statute 25 Charles II. c. 2; that the statute 1 William and Mary, st. 1. c. 9, for removing Papists, or reputed Papists, from the cities of London and Westminster, shall not extend to Roman Catholics taking the appointed oath; and that no peer of Great Britain or Ireland, taking that oath, shall be liable to be prosecuted for coming into his Majesty's presence, or into the court or house where his Majesty resides, under statute 30 Charles II. st. 2, c. 1. The fifth part of the act repeals the laws requir. ing the deeds and wills of Roman Catholics to be registered or inrolled; the sixth excuses persons acting as counsellors at law, barristers, attornies, clerks, or notaries, from taking the oath of supremacy, or the declaration against transubstantiation. But it is adviseable to take the oath of 18 George III. 30, to prevent all doubts, or ability to take by descent or purchase.

As the statute 1 William and Mary, st. 1, c. 18, called the Toleration Act, does not apply to Catholics, or persons denying the Trinity, they cannot serve in corporations, and are liable to the test and corporation act. They cannot sit in the House of Com mons, nor vote at elections, without taking the oath of supremacy; and cannot present to advowsons, although Jews and Quakers may. But the person is only disabled from presenting, and still continues patron. It seems they may serve on juries, but Catholic ministers are exempted. They also are entitled to attend the British factories and their meetings abroad, and may hold offices to be wholly exercised abroad, and may also serve under the East India Company, or in the army abroad, and the sixtieth regiment is chiefly composed of persons who cannot serve in England, by reason of the officers being many of them Catholics. This account of the state of the laws against Papists is extracted from an able review of them given by Mr. Butler, a Roman Catholic, in his Notes upon Lord Coke's Commentary on Littleton's Tenures, and which is to be found also in Tomlin's Law Dictionary, last edition, title PAPIST.

PAPPOPHORUM, in botany, a genus of the Triandria Digynia class and order. VOL. V.

Natural order of Gramina, or Grasses. Essential character: calyx two-valved, twoflowered; corolla two-valved, many awned. There is but one species; viz. P. alopecu roideum, a native of Spanish Town in America.

PAPPUS, in botany, thistle-down, a sort of feathery or hairy crown, with which many seeds, particularly those of compound flowers, are furnished for the purpose of dissemination. A seed surmounted by its pappus resembles a shuttle cock, so that it is naturally framed for flying, and for being transported by the wind to very considerable distances from its parent plant. By this contrivance of nature, the dandelion, groundsel, &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 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 shews what other nations should allow in exchange, which is rendered certain and fixed by the intrinsic value of the several species to be exchang. ed: but the course shews 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 Ex

CHANGE.

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, DEF, so that the side D E may be applied to the right line AB, and the other EF turned to the side on which the point C is situated. This done, and the thread FGC, 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, DE, along the right line A B, and by means of a pin, G, continually apply the thread to the side of the rule, EF, so as to keep it always stretched as the rule is moved along, the point of this pin, will describe a parabola GHO.

Definitions. 1. The right line A B is called the directrix. 2. The point C is the focus of the parabola. 3. All perpendiculars to the directrix, as LK, MO, &c. are called diameters; the points, where these cut the parabola, are called its vertices; the diameter BI, 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, GC, 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 EGG 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, ADC, 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 D E, and let F A, FC, and AC be joined; also let F G be drawn perpendicular to the directrix. Then, because (by Prop. 1), DA≈ DC, DF common to both, and the angle FDA FDC, FC will be equal to FA; but FA greater than FG therefore FC 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 HK a tangent in H. To demonstrate this, from any point K draw K L perpendicular to À B, and join K C; then because K C is greater than CH HB = KL, it follows that KC 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 E e, be parallel to the tangent DK; and let AD be a diameter passing through the point of contact D, and meeting E e in L; then shall EL= Le.

Let AD meet the directrix in A, and from the points E e, let perpendiculars EF, ef, be drawn to the directrix ; let CA be drawn, meeting Ee in G; and on the centre E, with the distance EC, let a circle be described, meeting AC again in H, and touching the directrix in F; and let DC be joined. Then because DA = DC, and the angle ADK = the angle CDK, it follows (4. 1.) that DK per. pendicular to AC; wherefore Ee perpendicular to A C, and CG GH (3. 3); so that e Ce H (4. 1), and a circle described upon the centre e, with the radius e C, must pass through H; and because e-C

ef, it must likewise pass through f. Now because Ff is a tangent to both these circles, and AHC cuts them, the square AF

the rectangle CA H (36. 3) = the square Af; therefore A FAƒ, and FE, AL, 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 HF, and the parameter of the axis, or to four times the rectangle HFB.

1. When the diameter is the axis; let DH be perpendicular B C, join DC, and draw DA perpendicular A B, and let F be the vertex of the axis. Then, because HB=DA=DC, it follows that HB =DC2=DH2 + HC2. Likewise, because BF FC, H B2 = 4 times the rectangle H FC+HC' (by 8. 2). Wherefore DH+H C2 4 times the rectangle HFB HC; and DH2 = 4 times the rectangle H FB; that is, DH' the rect angle contained under the absciss HF, and the parameter of the axis.

2. When the diameter is not the axis: let EN (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 EN 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 EF 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 CPK, CBA are right, and the angle B C P common, the triangles C BA, CPK are equiangular; and AC:CB (or CK: CP)::OK: GP; and ACXGP=OK × CB. Again, because CA = = 2CP, and CH2CG, AH= 2 GP; and consequently the rectangle CAH CA × 2 G P = OK × 2 CB. But, EN FA2 = rectangle CAH; and consequently, EN2 = OK × 2 CB = the rectangle contained under the absciss, LD, 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 another 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 EL, QR

be ordinates to the same diameter DN; and let EN, QS be perpendiculars to it. Then, on account of the equiangular triangles E LN, QRS, E L': Q R2 : : E N2 : QS2: that is, as the absciss DL to the absciss DR.

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

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

4 M Q2: wherefore Q R2 = 4 times the rectangle MQB + 4 M Q2; that is, to 4 times the rectangle QM B. But MQ= QK = DR, and M B = DA; wherefore Q R2 4 times the rectangle RDA: and because QR, EL are ordinates to the diameter A D, QR2 (by cor. 2, of Prop. 4), : EL (: RD LD): 4 times the rectangle RDA : 4 times the rectangle L D A. Therefore E L' 4 times the rectangle LDA, or the rectangle contained under the absciss L D, and the parameter of the diameter AD: 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, AC, to the diameter BC; 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 or dinate to the diameter A E; and the absciss BC will be equal to the absciss A E, or B D. Q. E. D.

Hence, if A C be an ordinate to BC, and AD be drawn so as to make B D = DC, then is AD 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 DC.

"To draw Tangents to the Parabola.” If the point of contact C be given: (fig. 7) draw the ordinate CB, and produce the axis till AT be AB; then join TC, which will be the tangent. Or if the point be given in the axis produced: take AB

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 D H 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 D C 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 AI 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 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 C B will be the tangent required.

"To find the Area of a Parabola." Multiply the base EG by the perpendicular height A I, and of the product will be the 季 area of the space A EGA; because the parabolic space is of its circumscribing parallelogram.

"To find the Length of the Curve AC," commencing at the vertex. Let y = the 2y

ordinate B C, p= the parameter, q ==,

and s=

√1+q'; then shall px (qs+ hyp. log. of qs) be the length of the

curve A C.

PARABOLA, Cartesian is a curve of the second order, expressed by the equation x y = ax3 + bx2 + cx+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 BN, MN 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 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 u x2 + bx + c, and an d

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 equa tion y y = a 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 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 differ ent 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 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 APB erect a prism, whose altitude is AS; this will be a parabolical cuneus, which of neces sity 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 parabolic pyramidoid.

The solidity hereof is had by multiplying the base by half the altitude, the reason of which is obvious; for the component 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.

PARABOLIC space, the area contained be tween any entire ordinate as V V (Plate

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