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 ordinate 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 rx y (m + r) to xy; that is, as r to mr.

PARABOLIC spindle, in guaging; a cask of the second variety is called the middle frustrum of a parabolic spindle. The parabolic spindle is eight fifteenths of its circum scribing 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 parading place appointed them at the head of some regiment.

PARADISEA, the bird of Paradise, in natural history, a genus of birds of the order Picæ. 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 enumerates twelve species, and Latham eight. P. apoda, or the greater Paradise bird, is about as large as a thrush. These birds are supposed to breed in North Guinea, whence they migrate into Aroo, returning 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 fea thers in sudden shiftings of the wind, and unable to proceed in their flight; are easily taken by the natives who also catch them with birdlime, and shoot them with blunted arrows. They are sold at Aroo for an iron nail each, and at Banda for half a rix-dollar.

Their food is not ascertained, and they cannot be kept alive in confinement. The smaller bird of Paradise is supposed by Latham to be a mere variety of the above. It is found only in the Papuan islands, where it is caught by the natives often by the hand, and exenterated and seared with a hot iron in the inside, and then put into the hollow of a bamboo to secure its plumage from injury.

PAKADOX, in philosophy, a proposi tion seemingly absurd, as being contrary to some received opinion; but yet true in fact. No science abounds more with paradoxes than geometry; thus, that a right line should continually approach to the hyperbola, and yet never reach it, is a true paradox; and in the same manner, a spira! may continually approach to a point, and yet not reach it, in any number of revolu tions, however great.

PARAGOGE, in grammar, a figure 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 CE A (Plate Parabola, &c. fig. 11) BTA, under which its true and apparent distance from the zenith is seen; or, which is the same thing, it is the angle TSE. The sines of the paral lactic angle A LT, A ST (fig. 12) at the same or equal distances, ZS, from the zenith, are in the reciprocal ratio of the distances TL, and TS, from the centre of the earth.

PARALLAX, in astronomy, denotes a change of the apparent place of any hea venly body, caused by being seen from different points of view; or it is the difference between the true and apparent distance of any heavenly body from the zenith. Thus let AB (Plate XII. Miscell. fig. 1) be a quadrant of a 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 VNH 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 TC. 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 DE; 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 pheno

menon.

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 A in the same line TAV, 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 star F is only GD, 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 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 PN is the parallax.

These distances are measured by the angles VT M, and VAM, but VA MVTM=TMA. For the external angle VAM angle ATM+ angle AM T, the two inward and opposite angles; so that AMT measures the parallax, and upon that account is itself frequently called the parallax; and this is always the angle under which the semi-diameter of the earth AT, 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 TAC, rectangular at A, having the semi-diameter of the earth, and TC the distance of the star, the angle A CT, 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 AT, and the angle ACT, the dis tance 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 sen→ sible 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 instant take the altitudes of both horns: the difference of these two altitudes being halved and added to the least, or subtracted from the greatest, gives nearly the visible or apparent altitude of the moon's centre; and the true altitude is nearly equal to the altitude of the centre of the shadow at that time. Now we know the altitude of the shadow, because we know the place of the sun in the ecliptic, and its depression under the horizon, which is equal to the altitude of the opposite point of the ecliptic in which is the centre of the shadow. And therefore having both the true altitude of the moon and the apparent altitude, the difference of these is the parallax required. But as the parallax of the moon increases as she approaches towards the earth, or the perigæum of her orbit; therefore astronomers have made tables, which shew the horizontal parallax for every de gree of its anomaly.

The parallax always diminishes the altitude of a phenomenon, or makes it appear lower than it would do, if viewed from the centre of the earth; and this change of the altitude may, according to the different situation of the ecliptic and equator in respect of the horizon of the spectator, cause a change of the latitude, longitude, declination and right ascension of any phenomenon, which is called their parallax. The pa rallax, therefore, increases the right and oblique ascension; diminishes the descension; diminishes the northern declination and latitude in the eastern part, and increases them in the western; but increases the southern both in the eastern and western part; diminishes the longitude in the western part, and increases it in the eastern. Hence it appears, that the parallax has just opposite effects to refraction. See REFRACTION,

PARALLAX, annual, the change of the apparent place of a heavenly body, which is caused by being viewed from the earth in different parts of its orbit round the sun. The annual parallax of all the planets is found very considerable, but that of the fixed stars is imperceptible.

PARALLAX, in levelling, denotes the angle contained between the line of the true level, and that of the apparent level.

PARALLEL. The subject of parallel lines, says Playfair, is one of the most difficult in the Elements of Geometry. It has accordingly been treated in a great variety of different ways, of which, perhaps, there is none which can be said to have given entire satisfaction. The difficulty consists in converting the twenty-seventh and twentyeighth of Enclid, or in demonstrating, that parallel straight lines (or such as do not meet one another) when they meet a third line, make the alternate angles with it equal, or which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to demonstrate this proposition, Euclid assumed it as an axiom, that if a straight line meet two straight lines, so as to make the interior angles on the same side of it less than two right angles, these straight lines being continually produced, will at length meet on the side on which the angles are that are less than two right angles. This proposition, however, is not self-evident; and ought the less to be received without proof, that the converse of it is a proposition that confessedly requires to be demonstrated. In order to remedy this defect, three sorts of methods have been adopteda new definition of parallel lines; a new manner of reasoning on the properties of straight lines without any new axiom; and, the introduction of a new axiom less exceptionable than Euclid's. Playfair adopts the latter plan; but we do not perceive that his axiom is by any means self-evident upon Euclid's definition which he retains, viz. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. A more intelligible, and we think an equally rigid, demonstration of the property of parallels, may be obtained without any axiom, by means of a new definition. It may at first sight be thought that the objection urged by Playfair against the definition in T. Simp. son's first edition, must equally hold against ours; but we think that if his objection really hold good against that definition, (though we confess we cannot feel the force

of it), it is obviated by distinguishing as ought to be done between the distance and the measure of that distance.

We must of course suppose our readers acquainted with the propositions in Euclid preceding the twenty-seventh; but to save the necessity of reference we shall give an enunciation of those which we shall have to employ in our demonstration, in the form in which we employ them. 1. (Prop. 16.) If one side of a triangle be produced, the outward angle is greater than either of the inward opposite angles. 2. (Prop. 19.) The greater angle of every triangle has the greater side opposite to it. 3. (Prop. 4.) If two triangles have two sides of the one respectively equal to two sides of the other, and have the included angles equal, the other angles will be respectively equal, viz. those to which the equal sides are opposite. 4. (Prop. 15.) If two straight lines cut each other, the vertical or opposite angles will be equal. 5. (Prop. 13.) If a straight line meet another, the sum of the adjacent angles is equal to the sum of two right angles.

6. Definition. Parallel straight lines are those whose least distances from each other are every where equal.

7. Theorem I. The perpendicular drawn to a straight line from any point, is the least line that can be drawn from that point to the given line.

Let C D, (Plate XII, Miscell, fig. 2) be a straight line drawn from C perpendicular to A B; and let C E be any other straight line from C to A B; then is CD less than CE. For the angle C D E equals angle CDA by construction; and CDA is greater than CED (1); therefore C D E is greater than CED. Hence (2) CD is less than CE.

8. Cor. 1. Hence the perpendicular from any point to a straight line is the true measure of the least distance of that point from that line.

9. Cor. 2. Hence (6) the perpendiculars to one of two parallel straight lines, from any points in the other, are every where equal to each other.

10. Cor. 3. Hence two parallel straight lines, however far they may be produced, can never meet.

11. Theorem II. If a line meeting two parallel straight lines be perpendicular to one of them, it is also perpendicular to the other.

If AB, (fig. 3) be parallel to CD, and EF meet them so as to be perpendicular to A B, it will also be perpendicular to CD. If not, draw E G perpendicular to CD and

parallel, and terminating with the poles. They are called parallels of latitude, because all places lying under the same pa rallel, have the same latitude.

PARALLELS of latitude, in astronomy, are lesser circles of the sphere parallel to the ecliptic, imagined to pass through every degree and minute of the colures. They are represented on the globe by the divisions on the quadrant of altitude, in its motion round the globe, when screwed over the pole of the ecliptic. See GLOBE.

PARALLELS of altitude, or ALMUCANTARS, are circles parallel to the horizon, imagined to pass through every degree and minute of the meridian between the horizon and zenith, having their poles in the zenith. They are represented on the globe by the divisions on the quadrant of altitude, in its motion about the body of the globe, when screwed to the zenith.

PARALLELS of declination, in astronomy, are the same with parallels of latitude in geography.

PARALLEL sphere, that situation of the sphere, wherein the equator coincides with the horizon, and the poles with the zenith and nadir. In this sphere all the parallels of the equator become parallels of the horizon, consequently, no stars ever rise or set, but all turn round in circles parallel to the horizon; and the sun when in the equinoctial, wheels round the horizon the whole day. After his rising to the elevated pole, he never sets for six months; and after his entering again on the other side of the line, never rises for six months longer. This is the position of the sphere to such as live under the poles, and to whom the sun is never above 23° 30′ high.

PARALLEL sailing, in navigation, is the sailing under a parallel of latitude. See NAVIGATION.

PARALLELEPIPED, or PARALLELOPIPED, in geometry, a regular solid comprehended under six parallelograms, the opposite ones whereof are similar, parallel, and equal. All parallelepipeds, prisms, cylinders, &c. whose bases and heights are equal, are themselves equal. A diagonal plane divides a parallelepiped into two equal prisms; so that a triangular prism is half a parallelepiped upon the same base, and of the same altitude.

All parallelepipeds, prisms, cylinders, &c. are in a ratio compounded of their bases and altitudes; wherefore, if their bases be equal, they are in proportion to their altitudes, and conversely. All parallelepipeds,

cylinders, cones, &c. are in a triplicate ratio, of their homologous sides, and also of their altitudes.

Equal parallelepipeds, prisms, cones, cylinders, &c. reciprocate their bases and altitudes.

PARALLELISM, the situation or quality whereby any thing is denominated parallel. See PARALLEL.

PARALLELISM of the earth's axis, in astronomy, that situation of the earth's axis, in its progress through its orbit, whereby it is still directed towards the pole-star; so that if a line be drawn parallel to its axis, while in any one position, the axis, in all other positions, will be always parallel to the same line.

This parallelism is the result of the earth's double motion, viz. round the sun, and round its own axis; or its annual and diurnal motion; and to it we owe the vicissitudes of seasons, and the inequality of day and night.

PARALLELISM of the rows of trees. These are never seen parallel, but always inclining to each other towards the further extreme. Hence mathematicians have taken occasion to inquire in what lines the trees must be disposed to correct this effect of the perspective, and make the rows still appear parallel. The two rows must be such, as that the unequal intervals of any two opposite or correspondent trees may be seen under equal visual rays.

PARALLELOGRAM, in geometry, a quadrilateral right-lined figure, whose opposite sides are parallel and equal to each other. It is generated by the equable motion of a right line always parallel to itself. When it has all its four angles right, and only its opposite sides equal, it is called a rectangle or oblong. When the angles are all right, and the sides equal, it is called a square. If all the sides are equal, and the angles unequal, it is called a rhombus or lozenge; and if the sides and angles be unequal, it is called a rhomboides.

In every parallelogram of what kind soever, a diagonal divides it into two equal parts; the angles diagonally opposite are equal; the opposite angles of the same side are together equal to two right angles; and each two sides, together, greater than the diagonal.

Two parallelograms on the same or equal base and of the same height, or between the same parallels, are equal; and hence two triangles on the same base and of the same height, are also equal. Hence,