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vided in o, so that the degrees in Co, op, be equal, and the perpendicular o S be erected to the line of measures gh. Then the line pn, Cl, drawn from the poles C, p, through any point Q in the line o S, will cut off the arches Fn, hl, equal to each other, and to the angle Q C p.

The great circle AO perpendicular to the plane of the primitive is projected into the straight line oS perpendicular to gh, by Prop. i. Cor. 3. Let Q be the projection of q; and since p Q, CQ, are straight lines, they are therefore the representations of the arches Pq, Cq, of great circles. Now, since PqC is an isosceles spherical triangle, the angles PCQ, CPQ, are therefore equal; and hence the arches Pq, Cq, produced will cut off equal arches from the given circles FI, GH, whose representations Fn, hl, are therefore equal: and, since the angle QC p is the measure of the arch hl, it is also the measure of its equal Fn.

COROL. Hence, if from the projected pole of any circle a perpendicular be erected to the line of measures, it will cut off a quadrant from the representation of that circle.

PROP. VIII. THEOR. VIII. Let. Fnk, fig. 2, be the projection of any circle FI, and p the projection of its pole P. If Cg be the cotangent of C A P, and g B perpendicular to the line of measures g C, let C A P be bisected by AO, and the line o B drawn to any point B, and also pB, cutting Fnk in d; then the angle go B is the measure of the arch Fd.

The arch PG is a quadrant, and the angle go A=gPA+oAP=gAC+0AP = FgAC +CAo=gAo; therefore gA=go; consequently o is the dividing centre of g B, the representation of GA; and hence by Prop. V. the angle go B is the measure of g B. But, since pg represents a quadrant, therefore p is the pole of gB; and hence the great circle pd B, passing through the pole of the circles g B and Fn, will cut off equal arches in both, that is, Fd=g B = angle go B.

COROL. The angle go B is the measure of the angle gp B. For the triangle gp B represents a triangle on the sphere, wherein the arch which g B represents is equal to the angle which the angle p represents; because gp is a quadrant; therefore go B is the measure of both.

PROP. IX. PROB. I. To draw a great circle through a given point, and whose distance from the pole of projection is equal to a given quantity.

Let AD B, fig. 3, be the projection, C its pole or centre, and P the point through which a great circle is to be drawn: through the points P, C, draw the straight line PCA, and draw CE perpendicular to it: make the angle CAE equal to the given distance of the circle from the pole of projection C; and from the centre C, with the radius C E, describe the circle EFG: through P draw the straight line PI K, touching the circle EFG in I, and it will be the projection of the great circle required.

PROP. X. PROB. II. To draw a great circle perpendicular to a great circle which passes through the pole of projection, and at a given distance from that pole.

Let AD B, fig. 3, be the primitive, and CI the given circle: draw C L perpendicular to CI,

and make the angle CLI equal to the given distance: then the straight line CP, drawn through I parallel to C L, will be the required projection.

PROP. XI. PROB. III. At a given point in a projected great circle, to draw another great circle to make a given angle with the former; and, conversely, to measure the angle contained between two great circles.

Let P, fig. 4, be the given point in the given great circle P B, and C the centre of the primitive: through the points P, C, draw the straight line POG, and draw the radius of the primitive CA perpendicular thereto; join PA; to which draw A G perpendicular: through G draw BGD at right angles to GP, meeting PB in B; bisect the angle CAP by the straight line AO; join BO, and make the angle BØD equal to that given; then, DP being joined, the angle BPD will be that required.

If the measure of the angle BPD be required, from the points B, D, draw the lines B O, DO, and the angle BOD is the measure of BPD.

PROP. XII. PROB. IV. To describe the projection of a less circle parallel to the plane of projection, and at a given distance from its pole.

Let A D B, fig. 3, be the primitive, and C its centre: set the distance of the circle from its pole, from B to H, and from H to D; and draw the straight line A E D, intersecting CE perpendicular to BC, in the point E: with the radius CE describe the circle EFG, and it is the projection required.

PROP. XIII. PROB. V. To draw a less circle perpendicular to the plane of projection.

Let C, fig. 5, be the centre of projection, and TI a great circle parallel to the proposed less circle: at C make the angles ICN, TC O, each equal to the distance of the less circle from its parallel great circle TI; let CL be the radius of projection, and from the extremity L draw L M perpendicular thereto; make C V equal to L M, or C F equal to C M; then, with the vertex V and assymptotes CN, CO, describe the hyperbola WV K; or, with the focus F and CV, describe the hyperbola, and it will be the perpendicular circle described.

PROP. XIV. PROB. VI. To describe the projection of a less circle inclined to the plane of projection.

Draw the line of measures dp, fig. 6, and at C, the centre of projection, draw CA perpendicular to dp, and equal to the radius of projection: with the centre A, and the radius AC, describe the circle DCF G; and draw RA E parallel to d p: then take the greatest and least distances of the circle from the pole of projection, and set them from C to D and F respectively; for the circle D F; and from A, the projecting point, draw the straight lines A Fƒ, and ADd; then df will be the transverse axis of the ellipse: but if D fall beyond the line R E, as at G, then from G draw the line GAD d, and dƒ is the transverse axis of an hyperbola: and if the point D fall in the line R E, as at E, then the line A E will not meet the line of measures and the circle will be projected into a parabola whose vertex is f: bisect df in H, the centre, and for the ellipse take half the difference of the lines A ḍ, A ƒ, which laid from H will give

K the focus; for the hyperbola, half the sum of Ad, Af being laid from H, will give k its focus: then with the transverse axis df, and focus K, or k, describe the ellipse d Mf, or hyperbola fm, which will be the projection of the inclined circle for the parabola, make EQ equal to Ff, and draw fn perpendicular to AQ, and make fk equal to one half of n Q: then with the vertex f, and focus k, describe the parabola fm, for the projection of the given circle F E.

PROP. XV. PROB. VII. To find the pole of a given projected circle.

Let D M F, fig. 7, be the given projected circle, whose line of measures is DF, and C the centre of projection; from C draw the radius of projection CA, perpendicular to the line of measures, and A will be the projecting point: join A D, AF, and bisect the angle DAF by the straight line AP; hence P is the pole. If the given projection be an hyperbola, the angle fAG, fig. 6, bisected, will give its pole in the line of measures; and, in a parabola, the angle fAE bisected will give its pole.

PROP. XVI. PROB. VIII. To measure any portion of a projected great circle, or to lay off any number of degrees thereon.

Let EP, fig. 8, be the great circle, and IP a portion thereof to be measured: draw ICD perpendicular to IP; let C be the centre, and CB the radius of projection, with which describe the circle EBD; make IA equal to IB; then A is the dividing centre of EP; hence, A P being joined, the angle IAP is the measure of the arch I P. Or, if IAP be made equal to any given angle, then IP is the correspondent arch of the projection.

PROP. XVII. PROB. IX. To measure any arch of a projected less circle, or to lay off any number of degrees on a given projected less

circle.

Let Fn, fig. 9, be the given less circle, and Pits pole: from the centre of projection C draw CA perpendicular to the line of measures GH, and equal to the radius of projection; join AP, and bisect the angle CAP by the straight line AO, to which draw AD perpendicular: describe the circle G/H, as far distant from the pole of projection C as the given circle is from its pole P; and through any given point n, in the projected circle Fn, draw Dnl, then H is the measure of the arch F n. Or let the measure be laid from H to l, and the line D joined will cut off F n equal thereto.

PROP. XVIII. PROB. X. To describe the gnomonic projection of a spherical triangle, when three sides are given; and to find the measures of either of its angles.

Let ABC, fig. 10, be a spherical triangle whose three sides are given: draw the radius CD, fig. 11, perpendicular to the diameter of the primitive EF; and at the point D make the angles CDA, CDG, ADI, equal respectively to the sides AC, BC, AB, of the spherical triangle ABC, fig. 10, the lines DA, DG, intersecting the diameter E F, produced if necessary in the points A and G; make DI equal to DG; then from the centre C, with the radius CG, describe an arch; and from A, with the distance A I, describe another arch, intersecting the for

mer in B; join AB, C B, and ACB will be the projection of the spherical triangle, and the rectilineal angle ACB is the measure of the spherical angle A C B, fig. 10.

PROP. XIX. PROB. XI. The three angles of a spherical triangle being given, to project it, and to find the measures of the sides.

Let A BC, fig. 12, be the spherical triangle of which the angles are given: construct another spherical triangle EFG, whose sides are the supplements of the given angles of the triangle ABC; and with the sides of this supplemental triangle describe the gnomonic projection, &c., as before. The supplemental triangle E F G has also a supplemental part E Fg; and when the sides G E, GF, which are substituted in place of the angles A, B, are obtuse, their supplements gE, g F, are to be used in the gnomonic projection of the triangle.

PROP. XX. PROB. XII. Given two sides, and the included angle of a spherical triangle, to describe the gnomonic projection of that triangle, and to find the measures of the other parts.

Let the sides AC, C B, and the angle AC B, fig. 10, be given: make the angles CDA, CDG, fig. 13, equal respectively to the sides A C, C B, fig. 10; also make the angle AC B, fig. 13, equal to the spherical angle AC B, fig. 10, and C B equal to CG, and ABC will be the projection of the spherical triangle.

To find the measure of the side A B: from C draw CL perpendicular to A B, and C M parallel thereto, meeting the circumference of the primitive in M; make LN equal to LM; join A N, BN, and the angle A BN will be the measure of the side A B. To find the measure of either of the spherical angles, as BAC from D draw DK perpendicular to A D, and make KH equal to K D: from K draw K I perpendicular to C K, and let A B produced meet K I in I, and join HI: then the rectilineal angle KHI is the measure of the spherical angle BAC. By proceeding in a similar manner, the measure of the other angle will be found.

PROP. XXI. PROB. XIII. Two angles and the intermediate side given, to describe the gnomonic projection of the triangle; and to find the measures of the remaining parts.

Let the angles CAB, AC B, and the side A C of the spherical triangle CDA, fig. 10, be given: make the angle CDA, fig. 13, equal to the measure of the given side A C, fig. 10; and the angle AC B, fig. 13, equal to the angle AC B, fig. 10, produce AC to H, draw D K perpendicular to C K, and make the angle KHI equal to the spherical angle CAB: from I, the intersection of K I, HI, to A draw IA, and let it intersect C B in B, and AC B, fig. 10. The unknown parts of this triangle may be measured by last problem.

PROP. XXII. PROB. XIV. Two sides of a spherical triangle, and an angle opposite to one of them given, to describe the projection of the triangle; and to find the measure of the remaining parts.

Let the sides A C, C B, and the angle BAC of the spherical triangle ABC, fig. 10, be given: make the angles CDA, CDG, fig. 13, equal respectively to the measures of the given sides

AC, BC: draw DK perpendicular to AD, make KH equal to DK, and the angle KHI equal to the given spherical angle BAC: draw the perpendicular KI, meeting HI in I; join AI; and from the centre C, with the distance CG, describe the arch G B, meeting AI in B; join C B, and A B C will be the rectilineal projection of the spherical triangle ABC, fig. 10; and the measures of the unknown parts of the triangle may be found as before.

PROP. XXIII. PROB. XV. Given two angles and a side opposite to one of them, to describe the gnomonic projection of the triangle, and to find the measures of the other parts.

Let the angles A, B, and the side BC of the triangle ABC, fig. 12, be given: let the supplemental triangle EFE be formed, in which the angles E, F, G, are the supplements of the sides BC, CA, A B, respectively, aud the sides EF, FG, GE, the supplements of the angles C, A, B. Now, at the centre C, fig. 13, make the angles CDA, CDK, equal to the measures of the sides GE, GF, respectively, being the supplements of the angles B and A; and let the lines DA, DK, intersect the diameter of the primitive EF, in the points A and K: draw DG perpendicular to A D, make G H equal to DG, and at the point H make the angle G H I equal to the angle E, or to its supplement; and let EI, perpendicular to CH, meet HI in I, and join AI: then from the centre C, with the distance CG, describe an arch intersecting A I in B; join CB, and A B C will be the gnomonic projection of the given triangle A BC, fig. 12: the supplement of the angle AC B, fig. 13, is the measure of the side AB, fig. 12; the measures of the other parts are found as before. Although this method of projection has, for the most part, been applied to dialling only, yet, from the preceding propositions, it appears that all the common problems of the sphere may be more easily resolved by this than by the ordinary methods of projection.

PROLATE, in geometry, is applied to a spheroid produced by the revolution of a semiellipsis about its larger diameter. See SPHE

ROID.

PROLEGOMENA, in philology, preparatory discourses fixed to a book, &c., containing something necessary to enable the reader the better to understand the book or science, &c. PROLEP'SIS, n. s. Į Fr. prolepse; Gr. PROLEPTICAL. Ο προληψις. A figure of rhetoric, in which objections are anticipated: in the manner of a prolepsis.

This was contained in my prolepsis or prevention of his answer. Bramhall against Hobbes. The proleptical notions of religion cannot be so well defended by the professed servants of the altar.

Glanville. Theobald.

This is a prolepsis or anachronism. PROLETA'RIAN, adj. Mean; wretched; vile; vulgar. A mean word whose etymology we do not find.

Like speculators should foresee,
From pharos of authority,
Portended mischiefs farther than
Low proletarian tything-men.

Hudibras.

PROLIFIC, adj. proles and facio.

Fr. prolifique; Lat.

PROLIF'ICAL.

Every dispute in religion grew prolifical, and in ventilating one question, many new ones were started. Decay of Piety. Main ocean flowed; not idle, but with warm Prolific humour soft'ning all her globe, Fermented the great mother to conceive, Satiate with genial moisture.

Milton's Paradise Lost. Their fruits, proceeding from simpler roots, are not so distinguishable as the offspring of sensible creatures, and prolifications descending from double origins. Browne.

His vital power air, earth, and seas supplies,
And breeds whate'er is bred beneath the skies;
For every kind, by thy prolific might,
Springs.

Dryden.

All dogs are of one species, they mingling together in generation, and the breed of such mixtures

PROIN, v. a. A corruption of prune. To being prolific. lop; cut; trim.

I sit and proin my wings

After flight, and put new strings To my shafts.

Ben Jonson.

The country husbandman will not give the proining knife to a young plant.

Id.

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From the middle of the world, The sun's prolific rays are hurled; 'Tis from that seat he darts those beams, Which quicken earth with genial flames.

PRO'LIX, adj. PROLIX IOUS,

Ray.

Prior. Fr. prolixe; Lat. prolixus. Long; tedious; verbose:

is a synonymc PROLIX LY, adv. Poined by Shakspeare pro lixity and prolixness, tediousness; tiresome dila

tion.

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Should I at large repeat

Prior.

The bead-roll of her vicious tricks,
My poem would be too prolix.
Elaborate and studied prolixity in proving such
points as nobody calls in question. Waterland.

PROLOCUTOR, n. s. Lat. prolocutor. The foreman; the speaker of a convocation.

The convocation the queen prorogued, though at the expence of Dr. Atterbury's displeasure, who was designed their prolocutor. Swift.

PROLOGUE, n. s. & v. d. Fr. prologue; Gr. póλoyos; Lat. prologus. Preface; introduction to a discourse or performance: to introduce with a preface.

Come, sit, and a song.

Shall we clap into 't roundly, without hawking. or spitting, or saying we are hoarse, which are the only prologues to a bad voice? Shakspeare.

If my death might make this island happy,
And prove the period of their tyranny,
I would expend it with all willingness;
But mine is made the prologue to their play. Id.
He his special nothing ever prologues.

In her face excuse

Id.

Came prologue, and apology too prompt. Milton.
From him who rears a poem lank and long,
To him who strains his all into a song;
Perhaps some bonny Caledonian air,

Prome; and there is here a royal menagerie of elephants. The ruins of the ancient city extend beyond the modern town, and contain a number of temples dedicated to Boodh. Long. 95° E., lat. 18° 50' N.

PROMETHEUS, the son of Japetus, supposed to have been the first discoverer of the art of striking fire by flint and steel; which gave rise to the fable of his stealing fire from heaven. This fable is variously related by different authors. Prometheus, as most say, being a man of subtle and crafty genius, in order to find out whether Jupiter was really worthy to be reckoned a god, slew two oxen, and stuffed one of the skins with the flesh, and the other with the bones of the The god, resolved to be revenged upon all manvictims, the latter of which was chosen by Jupiter. kind for this insult, deprived them of the use of fire; but Prometheus, with the assistance of Minerva, who had already aided him by her advice in forming the body of a man of tempered clay, contrived to ascend up to heaven, and, approaching the chariot of the sun, stole from thence the sacred fire, which he brought down to earth in a ferula. Jupiter, incensed at this strange and audacious enterprize, ordered Mercury to carry him to Mount Caucasus, and chain him to a rock, where an eagle was eternally to prey upon All birks and braes, though he was never there; Or, having whelped a prologue with great pains, his liver. This part of the history of PromeFeels himself spent, and fumbles for his brains; theus and his subsequent deliverance either by A prologue interdashed with many a strokeHercules or Jupiter himself, abounds with fieAn art contrived to advertise a joke, tions, which are supposed to contain some ancient So that the jest is clearly to be seen, facts under this disguise. M. Bannier supposes Not in the words-but in the gap between: that this is merely a continuation of the history of Manner is all in all, whate'er is writ, the Titans. Prometheus, as he conjectures, was The substitute for genius, sense, and wit. Cowper. not exempt from the persecutions which harassed PROLONG', v. a. Į Fr. prolonger; Lat. the other Titans. As he returned into Scythia, PROLONGA'TION, n. s. S pro and longus. To which he durst not quit so long as Jupiter lived, lengthen out; continue; draw out: hence, cor- that god is said to have bound him to Caucasus. ruptly, to put off a long time: prolongation is This prince, addicted to astrology, frequently rethe act of lengthening or delaying. tired to Mount Caucasus, as to a kind of observatory, where he contemplated the stars, and was, as it were, preyed upon by continual pining, or rather by vexation, on account of the solitary and melancholy life which he led. This is supposed to have given rise to the fable of the eagle or vulture that incessantly preyed upon his liver. Herodotus, however, alleges, that Prometheus was put in prison for not being able to stop the overflowing of a river, which from its rapidity obliged to fly with a part of his subjects to the was called the eagle, or at least that he was mountains to escape the inundation, till a traveller, represented by Hercules, undertook to dam it up by a mount, and to kill the eagle, as it may be said, by making its course regular and uniform; thus Prometheus was delivered by this hero from his prison, or retreat.

To-morrow in my judgment is too sudden;
For I myself am not so well provided,
As else I would be were the day prolonged.

tion of life.

Shakspeare.

Nourishment in living creatures is for the prolonga-
Bacon's Natural History.
This ambassage concerned only the prolongation of
days for payment of monies. Id. Henry VII.
Henceforth I fly not death, nor would prolong
Life much.

Milton.

The' unhappy queen with talk prolonged the night.
Dryden.

PROLU'SION, n. s.

Lat. prolusio. Entertainment; performance of diversion. It is memorable, which Famianus Strada, in the first book of his academical prolusions, relates of

Suarez.

Hakewill.

PROME, or PRONE, a city of the Birman empire, is situated on the eastern bank of the Irrawaddy, in a fine fertile plain, and was formerly surrounded by two walls, the exterior of timber, and the interior of brick. It is larger than Rangoon, and carries on a considerable trade in timber, grain, oil, wax, ivory, iron, lead, and flag-stones. It is said to have been once the capital of a dynasty. At present, with the adjoining territory, it forms the estate or appanage of one of the king's sons, called the prince of

Diodorus Siculus says that Prometheus first discovered combustible materials fit for kindling and maintaining fire. Bannier is of opinion, that the origin of this fiction was, that Jupiter, having ordered all the shops where iron was forged to be shut up, lest the Titans should make use of it against him, Prometheus, who had retired into Scythia, there established good forges; hence came the Calybes,' those excellent blacksmiths; and, perhaps Prometheus also, not thinking to find fire in that country, brought

some thither in the stalk of the ferula, in which it may be easily preserved for several days. As for the two oxen which Prometheus is said to

have slain, that he might impose upon Jupiter, this part of the fable is said to be founded upon his baving been the first who opened victims with a view of drawing omens from the inspection of their entrails. According to Le Clerc, Prometheus is the same with Magog, the former being the son of Japetus, and the latter the son of Japhet, and grandson of Noah. Both Prometheus and Magog settled in Scythia; the latter invented or improved the art of founding metals, and of forging iron, which the poets attributed to Prometheus; and Diodorus too says, that he invented several instruments for making fire. The appellation Magog signifies vexation, as Prometheus was gnawed by a vulture.

PROMETHEUS and DAMASICHTHON, two sons of Codrus, king of Athens, who conducted colonies

into Asia Minor.-Paus. i. c. 3.

PROMINENT, adj. Lat. prominens. PROMINENCE, or Standing out beyond PROMINENCY, N. S. another part; protuberant: the noun substantives both corresponding. Whales are described with two prominent spouts on their heads, whereas they have but one in the forehead, terminating over the windpipe.

Browne's Vulgar Errours. She has her eyes so prominent, and placed so that she can see better behind her than before her.

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are the servants of corruption. While they promise them liberty, they themselves 2 Peter ii. 13. As he promised in the law, he will shortly have mercy, and gather us together. 2 Mac. ii. 18. I eat the air, promise crammed; you cannot feed capons so. Shakspeare. His promises were, as he then was, mighty; But his performance, as he now is, nothing. Id. Your young prince Mamillius is a gentleman of the greatest promise. Id. Winter's Tale. Promising is the very air o' the' time: it opens the eyes of expectation: performance is ever the duller Shakspeare.

for his act.

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As the preceptive part enjoins the most exact virtue, so is it most advantageously enforced by the promissory, which is most exquisitely adapted to the same end. Decay of Piety. What God commands is good; what he promises is infallible. Bp. Hall. Whoever seeks the land of promise, shall find many lets. Id. He that brought us into this field, hath promised us victory. Id. Contemplations. If he receded from what he had promised, it would be such a disobligation to the prince that he would never forget it. Clarendon.

Nor was he obliged by oath to a strict observation of that which promissorily was unlawful. Browne. Duty still preceded promise, and strict endeavour only founded comfort.

Fell.

I could not expect such an effect as I found, which seldom reaches to the degree that is promised by the prescribers of any remedies.

Temple's Miscellanies. · Behold, she said, performed in every part My promise made; and Vulcan's laboured art. Dryden.

I dare promise for this play, that in the roughness of the numbers, which was so designed, you will see

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