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zled him exceedingly. One day, while sailing on the Thames, he observed that, every time the boat tacked, the direction of the wind, estimated by the direction of the vane, seemed to change. This suggested to him the case of his observed epicycle, and he found it an optical illusion, occasioned by a combination of the motion of light with the motion of his telescope while observing the polar stars. Thus he established an incontrovertible argument for the Coernican system, and immortalised his name by is discovery of the aberration of the stars. The doctor now engaged in a series of observations for ascertaining all the phenomena of this discovery. In the course of these, which were continued for twenty-eight years, he discovered another epicyclical motion of the pole of the heavens. He found that the pole described an epicycle whose diameter was about 18", having for its centre that point of the circle round the pole of the ecliptic in which the pole would have been found independent of this new motion: and that the period of this epicyclical motion was eighteen years and seven months. It struck him that this was precisely the period of the revolution of the nodes of the moon's orbit. Of these results he gave a brief account to lord Macclesfield, then president of the Royal Society. Mr. Machin, to whom he also communicated the observations, gave him in return a very neat mathematical hypothesis, by which the motion might be calculated. Let E (fig. 1.) be the pole of the ecliptic, and

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of this circle; and its place, in this circumference, depends on the place of the moon's ascending node. Draw EPF and GPL perpendicular to it; let G L be the colure of the equinoxes, and E F the colure of thc solstices. Dr. Bradley's observations showed that the pole was in A when the node was in L, the vernal equinox. If the node recede to H, the winter solstice, the node is in B. When the mode is in the autumnal equinox, at G, the pole is at C; and when the pole is in F, the summer solstice, the pole is in D. In all intermediate situations of the moon's ascending node the pole is in a point of the circumference A B C D, three signs or 9° more advanced. By comparing together a great number of observations, Dr. Bradley found that the mathematical theory, and the calculation depending on it, would correspond much better with the observations, if an ellipse were substituted for the circle A B C D, making the longer axis AC 18", and the shorter, BD, 16”. D'Alembert determined, by the physical theory of gravitation, the axes to be 18" and 13", 4. These observations, and this mathematical theory, must be considered as so many astronomical facts, and the methods of computing the places of all celestial phenomena must be drawn from them, agreeably to the universal practice of determining every point of the heavens by its longitude, latitude, right ascension, and declination. This equation of the pole's motion makes a change in the obliquity of the ecliptic. The inclination of the equator to the ecliptic is measur-. ed by the arch of a great circle intercepted between their poles. If the pole be in O, instead of P, it is plain that the obliquity is measured by E O instead of EP. If E P be considered as the mean obliquity of the ecliptic, it is augmented by 9" when the moon's ascending node is in the vernal equinox, and consequently the pole in A. It is, on the contrary, diminished 9" when the node is in the autumnal equinox, and the pole in C; and it is equal to the mean when the node is in the colure of the solstices. This change of the inclination of the earth's axis to the plane of the ecliptic was called the nutation of the axis by Sir Isaac Newton; who showed that a change of nearly a second must obtain in a year by the action of the sun on the F. parts of the terrestrial spheroid. But e did not attend to the change which would be made in this motion by the variation which obtains in the disturbing force of the moon, in consequence of the different obliquity of her action on the equator, arising from the motion of her own oblique orbit. It is this change which now goes by the name of nutation, and we owe its discovery entirely to Dr. Bradley. The general change of the position of the earth's axis has been termed deviation by modern astronomers. It is easy to ascertain the quantity of this change of obliquity. When the pole is in 0, the arch A DCO is equal to the node's longitude from the vernal equinox, and that P M is its cosine; and (on account of the smallness of A P in comparison of EP) PM may be taken for the change of the obliquity of the ecliptic. This is therefore = 9’ x cos. long. node, and is additive to the mean obliquity, while O is in the semicircle B A D, that is, while the longitude of

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the node is from nine signs to three signs ; but subtractive while the longitude of the node changes from three to nine signs. But the nutation changes also the longitudes and right ascensions of the stars and planets by changing the equinoctial points, and thus occasioning an equation in the precession of the equinoctial points. The great circle or meridian which passes through the poles of the ecliptic and equator is always the solstitial colure, and the equinoctial colure is at right angles to it: therefore when the pole is in Por in O, EP or E O is the solstitial colure. Let S be any fixed star or planet, and let S E be a meridian or circle of longitude; draw the circles of declination PS, OS, and the circles M’ EM", m Em', perpendicular to PE, O E. If the pole were in its mean place P, the equiocial points would be in the ecliptic meridian M’EM", or that meridian would pass through the intersections of the equator and ecliptic, and the angle M' ES would measure the longitude of the star S. But, when the pole is in O, the ecliptic meridian m Em' will pass through the equinoctial points. The equinoctial points must therefore be to the west of their mean place, and the equation of the precession must be additive to that precession; and the longitude of the star S will now be measured by the angle m ES, which, in the case here represented, is greater than its mean longitude. The difference, or the equation of longitude, arising from the mutation of the

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the sine of the angle C PO, which, by what has been already observed, is equal to the longitude of the node: Theorem O M is equal

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E 9 x sin, long node This equation is additive sin. obliq. eclip. to the mean longitude of the star when O is in the semicircle C B A, or while the ascending node is passing backwards from the vernal to the autumnal equinox; but it is subtractive from it while O is in the semicircle A DC, or while the node is passing backwards from the autumnal to the vernal equinox; or, to express it more briefly, the equation is subtractive from the mean longitude of the star while the ascending node is in the first six signs, and additive to it while the node is in the last six signs. This equation of longitude is the same for all the stars; for their longitude is reckoned on the ecliptic, and therefore is affected only by the variation of the point from which the longitude is computed. The right ascension, being comso on the equator, suffers a double change. t is computed from, or begins at, a different point of the equator, and it terminates at a different point; because, the equator having changed its position, the circles of declination also change theirs. When the pole is at P the right ascension of S from the solstitial colure is measured by the angle S P F, contained between that colure and the star's circle of declination. But, when the pole is at O, the right ascension is measured by the angle SOF, and the difference of S P E and SOE is the equation of right as

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cension. The angle SOE consists of two parts, GO E and G O S ; G O E remains the same wherever the star S is placed, but G O S varies with the place of the star.—We must first find the variation by which G P E becomes G O E, which variation is common to all the stars. The triangles G PE, G O E, have a constant side GE, and a constant angle G; the variation PO of the side G P is extremely small, and therefore the variation of the angles may be computed by Mr. Cotes's Fluxionary Theorems. See Simpson's Fluxions, sect. 253, &c. As the tangent of the side EP, opposite to the constant angle G, is to the sine of the angle GPE, opposite to the constant side EG, so is PO the variation of the side GP, adjacent to the constant angle, to the variation r of the angle GPO, opposite to the con

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This is subtractive from the mean right ascension for the first six signs of the node's longitude, and additive for the last six signs. This equation is common to all the stars. We may discover the variation of the other part SOG of the angle, which depends on the different position of the hour circles P S and OS, which causes them to cut the equation in different points, where the arches of right ascension terminate, as follows:—The triangles SPG, S O G, have a constant side SG, and a constant angle G. Therefore, by the same Cotesian theorem, tan. S P : sin. S P G = PO: y, and y, or the second part of the nutation in right as9" x sin. diff. R. A. of star and node

cotan. declin. star.

The nutation also affects the declination of the stars: For SP, the mean codeclination, is changed into S.O.-Suppose a circle described round S, with the distanceSO cutting S P in f; then it is evident that the equation ofdeclin. is Pf- PO x cos. O Pf- 9’ x sign R. A. of star—long. of node.

These are the calculations constantly used in our astronomical researches, founded on Machin's Theory. When still greater accuracy is required, the elliptical theory must be substituted, by taking (as is expressed by the dotted lines) O in that point of the ellipse described on the transverse axis AC, where it is cut by OM, drawn according to Machin's theory. All the change made here is the diminution of O M in the ratio of 18 to 13:4, and a corresponding diminution of the angle C PO. The detail of it may be seen in De la Lande's Astronomy, art. 2874. The calculations being in every case tedious, and liable to mistakes, on account of the changes of the signs of the different equations, the zealous promoters of astronomy have calculated and published tables of these equations. We may now consider the precession of the equinoctial points, with its equations, arising from the mutation of the earth's axis, as a physical phenomenon, and endeavour to account for it upon those mechanical principles which have so happily explained all the other phenomena of the célestial motions. Sir Isaac Newton quickly found it to be a consequence, and the most beautiful proof, of the universal gravitation of matter. }. is no part of his immortal work

cension, E

where his sagacity and fertility of resource shine more conspicuously than in this investigation. His investigation, however, was only a shrewd guess, founded on assumptions, of which it would be extremely difficult to demonstrate either the truth or falsity, and which required the genius of a Newton to select in such a complication of abstruse circumstances. The subject has occupied the attention of the first mathematicians of Europe since his time; and is still considered as the most curious and difficult of mechanical problems. The most elaborate and accurate dissertations on the precession of the equinoxes are those of Sylvabella and Walmesly, in the Philosophical Transactions, published about 1754; that of Thomas Simpson, in his Miscellaneous Tracts; that of Frisius, in the Mem. of the Berlin Academy, and afterwards in his Cosmographia; that of Euler in the Memoirs of Berlin; that of D'Alembert in a separate dissertation; and that of de la Grange on the Libration of the Moon, which obtained the §. in the Academy of Paris in 1769. The issertation of Frisius is thought the most perspicuous of them all, being conducted in the meod of geometrical analysis; whereas most of the others proceed in the fluxionary and symbolic method, which does not give the same perspicuous conviction of the truth of the results. We shall here give a short sketch of Newton's investigation. Let S (fig. 2) be the sun, E the

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eartn, and M the moon, moving in the orbit N M C Dn, which cuts the plane of the ecliptic, in the line of the nodes Nn, and has one-half raised above it, as represented in the figure, the other half being hid below the ecliptic. Suppose this orbit folded down; it will coincide with the ecliptic in the circle N m can. Let EX represent the axis of this orbit, perpendicular to its go and therefore inclined to the ecliptic.

ince the moon gravitates to the sun in the direction MS, which is all above the ecliptic, it is plain that this gravitation has a tendency to draw the moon towards the ecliptic. Suppose this force to be such that it would draw the moon down from M to i in the time that she would have moved from M to t, in the tangent to her orbit. By the combination of these motions the moon will desert her orbit, and describe the line M r, which makes the diagonal of the

parallelogram; and, if no farther action of the sun be supposed, she will describe another orbit Mön', lying between the orbit M C D n and the ecliptic, and she will come to the ecliptic, and pass through it in a point n nearer to M than n is, which was the former place of her descending node. By this change of orbit, the line E X will no longer be perpendicular to it; but there will be another fine Er which will now be perpendicular to the new orbit. Also the moon, moving from M to r, does not move as if she had come from the ascending node N, but from a point N lying beyond it; and the line of the orbit in this new position is N'n'. Also the angle M N m is less than the angle M N m. Thus the nodes shift their places in a direction opposite to that of her motion, or move to the west; the axis of the orbit changes its position, and the orbit itself changes its inclination to the ecliptic. These momentary changes are different in different parts of the orbit, according to the position of the line of the nodes. Sometimes the inclination of the orbit is increased, and sometimes the nodes move to the east. But, in general, the inclination increases from the time that the nodes are in the line of syzigee, till they get into quadrature, after which it diminishes till the nodes are again in syzigee. The nodes advance only while they are in the octants after the quadrature, and while the moon passes from the quadrature to the node, and they recede in all other situations. Therefore the recess exceeds the advance in every revolution of the moon round the earth, and, on the whole, they recede. What has been said of one moon would be true of each of a continued ring of moons surrounding the earth, and they would thus compose a flexible ring, which would never be flat, but waved, according to the difference (both in kind and degree), of the disturbing forces acting on its different parts. But suppose these moons to cohere, and to form a rigid and flat ring, nothing would remain in this ring but the excess of the contrary tendencies of its different parts. Its axis would be perpendicular to its plane, and its position in any moment will be the mean position of all the axes of the orbits of each part of the flexible ring. Suppose this ring to contract in dimensions, the disturbing forces will diminish in the same proportion, and in this proportion. will all their effects diminish. Suppose its motion of revolution to accelerate, or the time of a revolution to diminish; the linear effects of the disturbing forces being as the square of the times of their action, and their angular effects as the times, those errors must diminish also on this account; and we can compute what those errors will be for any diameter of the ring, and for any period of its revolution. We can tell, therefore, what would be the motion of the nodes, the change of inclination, and deviation of the axis, of a ring which would touch the surface of the earth, and revolve in twenty-four hours; nay, we can tell what these motions would be, should this ring adhere to the earth. They must be much less than if the ring were detached. For the disturbing forces of the ring must drag along with it the whole globe of the earth. The quantity of motion which the disturbing forces would have produced in the ring alone, will now, says Newton, be produced in the whole mass; and therefore the velocity must be as much less as the quantity of matter is greaten: but still all this can be computed. That there is such a ring on the earth is certain; for the earth is not a sphere, but an elliptical spheroid. Sir Isaac Newton, therefore, made a computation of the effects of the disturbing force, and has exhibited a most beautiful example of mathematical investigation. He first asserts that the earth must be an elliptical spheroid, whose polar axis is to its equatorial diameter as 229 to 230. Then he demonstrates that if the sine of the inclination of the equator be called r, and if t be the number of days (sidereal) in a year, the annual motion of a detached ring will

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3 - 2. be 360° x 2x+2. He then shows that

the effect of the disturbing force on this ring is to its effect on the matter of the same ring, distributed in the form of an elliptical stratum (but still detached) as 5 to 2; therefore the motion 2 of the nodes will be 360° x *Hz. 16" 24” annually. He then proceeds to show that the quantity of motion in the sphere is to that in the equatorial ring revolving in the same time, as the matter in the sphere to the matter in the ring, and as three times the square of a quadrantal arch to two squares of a diameter, jointly: then he shows that the quantity of matter in the terrestrial sphere is to that in the protuberant matter of the spheroid as 52906 to 461 (supposing all homogeneous). From these premises it follows that the motion of 16' 16" 24" must be diminished in the ratio of 10717 to 100, which reduces it to 9'07" annually. And this, he says, is the precession of the equinoxes, occasioned by the action of the sun; and the rest of the 504", which is the observed precession, is owing to the action of the moon nearly five times greater than that of the sun. This appeared a great difficulty; for the phenomena of the tides show that it cannot much exceed twice the sun's force. The ingenuity of this process is justly celebrated by Daniel Bernouilli, who (in his Dissertation on the Tides, which shared the prize of the French Academy with M'Laurin and Euler) says that Newton saw through a veil what others could hardly discover with a microscope in the light of the meridian sun. His determination of the form and dimensions of the earth, which is the foundation of the whole process, is not offered as any thing better than a probable guess, in re difficillima; and it has been since demonstrated with geometrical rigor by M'Laurin. His next

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principle, that the motion of the nodes of the rigid ring is equal to the mean motion of the nodes of the moon, has been most critically discussed by the first mathematicians, as a thing which could neither be proved nor refuted. Frisius has at last shown it to be a mistake, and that the motion of the nodes of the ring is double the mean motion of the nodes of a single moon; and that Newton's own principles should have produced a precession o eighteen seconds and a quarter annually; which removes the difficulty formerly mentioned. Sir Isaac Newton's third assumption, that the quantity of motion of the ring must be shared with the included sphere, was acquiesced in by all his commentators, till D'Alembert and Euler, in 1749, showed that it was not the quantity of motion round an axis of rotation which remained the same, but the quantity of momentum or rotatory effort. The quantity of motion is the o of every particle by its velocity; that is, y its distance from the axis; while its momentum, or power of producing rotation, is as the square of that distance, and is to be had by taking the sum of each particle multiplied by the square of its distance from the axis. Since the earth differs so little from a perfect sphere, this makes no sensible difference in the result. It will increase Newton's precession about threefourths of a second. The source of Newton's mistake in the solution of this intricate problem was first detected by Mr. Landen, in the first volume of his Memoirs. That superior mathematician discovered that when a rigid annulus revolves with two motions, one in its own plane, and the other round one of its diameters, half the motive force acting upon the ring is counteracted by the centrifugal force arising from the compound motion, and half only is efficacious or accelerating the plane of the annulus round its diameter. Mr. Landen did not expressly demonstrate this; but it has been done very completely by Dr. o in the seventh volume of the Memoirs of the lish Academy. We cannot here pursue this subject; but beg to refer the reader to Dr. Milner's paper in the Philosophical Transactions; to Dr. Abram Robertson's paper in the Philosophical Transactions for 1807; to the Dissertation of Frisius already specified; and to the popular view of this problem by M. Laplace in his Exposition, book iv. ch. 13. To find the precession in right ascension and declination.—Put dic the declination of a star, and a = its right ascension; then their annual variations of precessions will be nearly as follow, viz. 20" .084 x cos. a = the annual precession in declinat, and 46" "O619 + 20"-084 x sin. a x tang. d = that of right ascension.

PRECIAE, precius, early, the twenty-first order in Linnaeus's fragments of a natural method; consisting of primrose, an early flowering ant, and a few genera which agree with it in it and structure. Sea BotANY.

PRECINCT, n. s. Lat. pracinctus. Out. ward limit; boundary.

The main body of the sea being one, yet within divers precincts, hath divers names; so the catholick church is in like sort divided into a number of distinct societies. Hooker.

This is the manner of God's dealing with those that have lived within the precincts of the church; they shall be condemned for the very want of true faith and repentance. Perkins.

Through all restraint broke loose, he wings his wa. Nor far oiewen. in the precincts of light, Directly towards the new created world. Milton. To find our hearthstone turned into a tomb, And round its once warm precincts palely lying The ashes of our hopes, is a deep grief,

Beyond a single gentleman's belief. Byron. PRECIOUS, adj. R Fr. precieur ; Latin PRE'ciously, adv. Valuable;

pretiosus. PRE'croUSNESS, n.s. $ of great worth; costly; often used in irony: the adverb and noun-substantive follow the senses of the adjective. A womman that hadde a boxe of alabastre of precious oynement cam to him and schedde out on the heed of him restynge. Wiclif. Matt. 26. The lips of knowledge are a precious jewel. Prov. xx. 15. Many things which are most precious, are neglected only because the value of them lieth hid. Hooker. I never saw Such precious deeds in one that promised nought But begg'ry and poor luck. Shakspeare. Cymbeline, Its preciousness equalled the price of pearls. Wilkins. Let none admire That riches grow in hell; that soil may best Deserve the precious bane. Milton. The index or forefinger was too naked whereto to commit their preciosities, and hath the tuition of the thumb scarce unto the second joint. Browne. Barbarians seem to exceed them in the curiosity of their application of these preciosities. More. Fortune, conscious of your destiny, Ev’n then took care to lay you softly by ; And wrapp'd your fate among her precious things, Kept fresh to be unfolded with your king's. Dryden. More of the same kind, concerning these precious saints amongst the Turks, may be seen in Pietro della Valle. - Loche. These virtues are the hidden beauties of a soul which make it lovely and precious in his sight, from whom no secrets are concealed. Addison's Spectator. PRECIPICE, n.s. Fr. precipice; Lat. pracipitium. A headlong or perpendicular steep. I ere long that precipice must tread, Whence none return that leads unto the dead. Sandys. You take a precipice for no leap of danger, And woo your own destruction. Shakspeare. Where the water dasheth more against the bottom, there it moveth more swiftly and more in precipice; for in the breaking of the waves there is ever a precipice. Bacon. No stupendous precipice denies Access, no horror turns away our eyes. Denham. Swift down the precipice of time it goes, And sinks in minutes, which in ages rose. Dryden. Drink as much as you can get; because a good coachman never drives so well as when he is drunk; and then shew your skill, by driving to an inch by a

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tant is, falling or rushing headlong; hasty; hurried: the adverb corresponding: precipitate, to urge or throw headlong; urge on with violence; hasten; hurry blindly; throw to the bottom by a chemical process: as a verb neuter, to fall headlong; fall to the bottom: precipitate as an adjective is synonymous with precipitant; as a noun substantive, it is a medical term for the red oxide of mercury: precipitately and precipitation correspond with the adjective: precipitous is, steep ; headlong; hasty; rash. Hadst thou been aught but goss'mer feathers, So many fathom down precipitating, Thou'dst shiver like an egg. Shakspeare. King Lear. Let them pile ten hills on the Tarpeian rock, That the precipitation might down-stretch Below the beam of sight, yet will I still Be this to them. Id. Coriolanus. Barcephas saith, it was necessary this paradise should be set at such a height, because the four rivers, had they not fallen so precipitate, could not have had sufficient force to thrust themselves under the great ocean. Raleigh. She had a king to her son-in-law, yet was, upon dark and unknown reasons, precipitated and banished the world into a nunnery. Bacon. As for having them obnoxious to ruin, if they be of fearful natures, it may do well; but, if they be daring, it may precipitate their designs, and prove dangerous. Id. By strong water every metal will precipitate. Id. Separation is wrought by precipitation or sublimation; that is, a calling of the parts up or down, which is a kind of attraction. - Id. The commotions in Ireland were so sudden and so violent, that it was hard to discern the rise, or apply a remedy to that precipitant rebellion. King Charles. Monarchy, together with me, could not but be dashed in pieces by such a precipitous fall as they intended. Id. Short intermittent and swift recurrent pains do precipitate patients into consumptions. arvey. They were wont, upon a superstition, to precipitate a man from some high cliff into the sea, tying about him with strings many great fowls. Wilkins. Dear Erythraea, let not such blind fury Precipitate your thoughts, nor set them working, Till time shall lend them better means Than lost complaints. Denham's Sophy. The archbishop, too precipitate in pressing the reception of that which he thought a reformation, paid dearly for it. Clarendon. Thither they haste with glad precipitance. Milton. Without longer pause, Downright into the world's first region throws His flight precipitant. Id. Paradise Lost. As the chymist, by catching at it too soon, lost the philosophical elixir, so precipitancy of our understanding is an occasion of error. Glanville. Though the attempts of some have been precipitous, and their enquiries so audacious as to have lost themselves in attempts above humanity, yet have the enquiries of most defected by the way. Browne's Vulgar Errours. The goddess guides her son, and turns him from the light, Herself involved in clouds, precipitates her flight. Dryden. Thus framed for ill, he loosed our triple hold, Advice unsafe, precipitous, and bold. Id. How precious the time is, how precipitous the oc

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