zled him exceedingly. One day, while sailing of this circle; and its place, in this circumferon the Thames, he observed that, every time the ence, depends on the place of the moon's ascendboat tacked, the direction of the wind, esti- ing node. Draw EPF and GPL perpendicular mated by the direction of the vane, seemed to to it; let G L be the colure of the equinoxes, change. This suggested to him the case of his and E F the colure of the solstices. Dr. Bradobserved epicycle, and he found it an optical il- ley's observations showed that the pole was in A lusion, occasioned by a combination of the mo when the node was in L, the vernal equinox. If tion of light with the motion of his telescope the node recede to H, the winter solstice, the while observing the polar stars. Thus he estab- node is in B. When the node is in the autumnal lished an incontrovertible argument for the Co- equinox, at G, the pole is at C; and when the pernican system, and immortalised his name by pole is in F, the summer solstice, the pole is in his discovery of the aberration of the stars. The D. In all intermediate situations of the moon's doctor now engaged in a series of observations ascending node the pole is in a point of the cirfor ascertaining all the phenomena of this disco- cumference A BCD, three signs or go more very. In the course of these, which were conti- advanced. By comparing together a great pumnued for twenty-eight years, he discovered ano- ber of observations, Dr. Bradley found that the ther epicyclical motion of the pole of the heavens. mathematical theory, and the calculation deHe found that the pole described an epicycle pending on it, would correspond much better whose diameter was about 18", having for its with the observations, if an ellipse were substicentre that point of the circle round the pole of tuted for the circle ABCD, making the longer the ecliptic in which the pole would have been axis AC 18", and the shorter, BD, 16". D'Alemfound independent of this new motion : and that bert' determined, by the physical theory of gravithe period of this epicyclical motion was eighteen tation, the axes to be 18" and 13“; 4. These years and seven months. It struck him that observations, and this mathematical theory, must this was precisely the period of the revolution of be considered as so many astronomical facts, the nodes of the moon's orbit. Of these results and the methods of computing the places of all he gave a brief account to lord Macclesfield, celestial phenomena must be drawn from them, then president of the Royal Society. Mr. agreeably to the universal practice of determinMachin, to whom he also communicated the ob- ing every point of the heavens by its longitude, servations, gave him in return a very neat ma- latitude, right ascension, and declination. thematical hypothesis, by which the motion might This equation of the pole's motion makes a be calculated. change in the obliquity of the ecliptic. The inLet E (fig. 1.) be the pole of the ecliptic, and clination of the equator to the ecliptic is measur-. ed by the arch of a great circle intercepted te tween their poles. If the pole be in 0, instead S of P, it is plain that the obliquity is measured by E O instead of EP. If EP 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 F 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. M This change of the inclination of the earth's axis G L to the plane of the ecliptic was called the nutaP tion 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 prominent parts of the terrestrial spheroid. But he did not attend to the change which would be made in this motion by the variation which obM' tains in the disturbing force of the moon, in conM' sequence 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 ADCO is equal to the node's longitude SP Q a circle distant from it 23° 28', represent- from the vernal equinox, and that P M is its coing the circle described by the pole of the equator sine ; and (on account of the smallness of AP during one revolution of the equinoctial points. in comparison of EP) PM may be taken for Let P be the place of this last mentioned pole at the change of the obliquity of the ecliptic. This some given time. Round P describe a circle, is therefore = 9' X cos. long. node, and is adABC D, whose diameter AC is 18'. The real ditive to the mean obliquity, while 0 is in the situation of the pole will be in the circumference semicircle B AD, that is, while the longitude of fig. 1. m' Q H the node is from nine signs to three signs ; but cension. The angle S O E consists of two parts, subtractive while the longitude of the node GO E and GOS; GO E remains the same changes from three to nine signs. But the nu- wherever the star S is placed, but GOS varies tation changes also the longitudes and right as with the place of the star.- We must first find censions of the stars and planets by changing the the variation by which G PE becomes GOE, equinoctial points, and thus occasioning an equa- which variation is common to all the stars. The tion in the precession of the equinoctial points. triangles G PE, GOE, have a constant side The great circle or meridian which passes through G E, and a constant angle G; the variation PO the poles of the ecliptic and equator is always of the side G P is extremely small, and therefore the solstitial colure, and the equinoctial colure is the variation of the angles may be computed by at right angles to it: therefore when the pole is in Mr. Cotes's Fluxionary Theorems. See SimpPor in 0, E P or E O is the solstitial colure. 'son's Fluxions, sect. 253, &c. As the tangent Let S be any fixed star or planet, and let S E be of the side EP, opposite to the constant angle a meridian or circle of longitude ; draw the cir- G, is to the sine of the angle GP E, opposite to cles of declination PS, OS, and the circles the constant side E G, so is P() the variation of M'EM", m Em', perpendicular to P E, O E. the side GP, adjacent to the constant angle, to the If the pole were in its mean place P, the equi- variation r of the angle GPO, opposite to the connoctial points would be in the ecliptic meridian M'EM", or that meridian would pass through 9" x sin.long. node stant side EG. This gives x= tang. obl. eclip. the intersections of the equator and ecliptic, and This is subtractive from the mean right ascenthe angle M'ES would measure the longitude of sion for the first six signs of the node's longitude, the star S. But, when the pole is in O, the eclip- and additive for the last six signs. This equatic meridjan m E m' will pass through the equi- tion is common to all the stars. noctial points. The equinoctial points must there We may discover the variation of the other fore be to the west of their mean place, and the part SO G of the angle, which depends on the equation of the precession must be additive to different position of the hour circles PS and that precession; and the longitude of the star SOS, which causes them to cut the equation in will now be measured by the angle m E S, which, different points, where the arches of right ascenin the case here represented, is greater than its sion terminate, as follows :—The triangles SPG, mean longitude. The difference, or the equation SOG, have a constant side SG, and a constant of longitude, arising from the nutation of the angle G. Therefore, by the same Cotesian OM earth's axis, is the angle O EP, or OM is theorem, tan. SP: sin. SPG=PO: y, and OE y, or the second part of the nutation in right asthe side of the angle CPO, which, by what 9" x sin. diff. R. A. of star and node cension, has been already observed, is equal to the cotan. declin, star. longitude of the node: Theorem O M is equal The nutation also affects the declination of the OM to que x longitude node, and is equal to stars: For SP, the mean codeclination, is changed OE into $ 0.-Suppose a circle described rounds, 9' X sin. long. node with the distance S O cutting S Pin f; then it is eviThis equation is additive dent that the equation of declin.is Pf=PO X cos. sin, obliq. eclip. to the mean longitude of the star when 0 is in OPf=9' x sign R. A. of star—long. of node. the semicircle C BA, or while the ascending These are the calculations constantly used in node is passing backwards from the vernal to the our astronomical researches, founded on Machin's autumnal equinox; but it is subtractive from it Theory: When still greater accuracy is required, while O is in the semicircle ADC, or while the the elliptical theory must be substituted, hy node is passing backwards from the autumnal to taking (as is expressed by the dotted lines) O in the vernal equinox; or, to express it more briefly, that point of the ellipse described on the transthe equation is subtractive from the mean longi- verse axis A C, where it is cut by O M, drawn tude of the star while the ascending node is in according to Machin's theory. All the change the first six signs, and additive to it while the made here is the diminution of O M in the ratio node is in the last six signs. of 18 to 13:4, and a corresponding diminution This equation of longitude is the same for all of the angle CPO. The detail of it may be the stars; for their longitude is reckoned on the seen in De la Lande's Astronomy, art. 2874. The ecliptic, and therefore is affected only by the calculations being in every case tedious, and variation of the point from which the longitude liable to mistakes, on account of the changes of the is computed. The right ascension, being com- signs of the different equations, the zealous proputed on the equator, suffers a double change. moters of astronomy have calculated and pubIt is computed from, or begins at, a different lished tables of these equations. point of the equator, and it terminates at a dif We may now consider the precession of the ferent point; because, the equator having chang- equinoctial points, with its equations, arising ed its position, the circles of declination also from the nutation of the earth's axis, as a physichange theirs. When the pole is at P the right cal phenomenon, and endeavour to account for ascension of S from the solstitial colure is mea- it upon those mechanical principles which have sured by the angle SP E, contained between so happily explained all the other phenomena of that colure and the star's circle of declination, the celestial motions. Sir Isaac Newton quickly But, when the pole is at O, the right ascension is found it to be a consequence, and the most measured by the angle SO E, and the difference beautiful proof, of the universal gravitation of of SPE and SO E is the equation of right as matter. There is no part of his immortal work where his sagacity and fertility of resource shine parallelogram; and, if no farther action of the more conspicuously than in this investigation. sun be supposed, she will describe another orbit His investigation, however, was only a shrewd Mdn', lying between the orbit MC Dn and the guess, founded on assumptions, of which it ecliptic, and she will come to the ecliptic, and would be extremely difficult to demonstrate pass through it in a point n' nearer to M than n either the truth or falsity, and which required the is, which was the former place of her descending genius of a Newton to select in such a compli- node. By this change of orbit, the line E X will cation of abstruse circumstances. The subject no longer be perpendicular to it; but there will has occupied the attention of the first mathema- be another fine Er which will now be perpenticians of Europe since his time; and is still dicular to the new orbit. Also the moon, moving considered as the most curious and difficult of from M to r, does not move as if she had come . mechanical problems. The most elaborate and from the ascending node N, but from a point N accurate dissertations on the precession of the lying beyond it; and the line of the orbit in this equinoxes are those of Sylvabella and Walmesly, new position is N'n'. Also the angle M N'm is in the Philosophical Transactions, published less than the angle M Nm. Thus the nodes shift about 1754; that of Thomas Simpson, in his their places in a direction opposite to that of her Miscellaneous Tracts; that of Frisius, in the motion, or move to the west; the axis of the Mem. of the Berlin Academy, and afterwards orbit changes its position, and the orbit itself in his Cosmographia ; that of Euler in the Me- changes its inclination to the ecliptic. These moirs of Berlin; that of D'Alembert in a sepa- momentary changes are different in different parts rate dissertation ; and that of de la Grange on of the orbit , according to the position of the line the Libration of the Moon, which obtained the of the nodes. Sometimes the inclination of the prize in the Academy of Paris in 1769. The orbit is increased, and sometimes the nodes move dissertation of Frisius is thought the most per- to the east. But, in general, the inclination inspicuous of them all, being conducted in the me creases from the time that the nodes are in the thod of geometrical analysis ; whereas most of line of syzigee, till they get into quadrature, after the others proceed in the Auxionary and symbolic which it diminishes till the nodes are again in method, which does not give the same perspicu- syzigee. The nodes advance only while they ous conviction of the truth of the results. are in the octants after the quadrature, and We shall here give a short sketch of Newton's while the moon passes from the quadrature to investigation. Let S (fig. 2) be the sun, E the the node, and they recede in all other situations. Therefore the recess exceeds the advance in Fig. 2. every revolution of the moon round the earth, X:30 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 řexible ring, which would never be fat, 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 moeartn, and M the moon, moving in the orbit NM tion of revolution to accelerate, or the time of a C Dn, which cuts the plane of the ecliptic, in revolution to diminish; the linear effects of the the line of the nodes Nn, and has one-half raised disturbing forces being as the square of the above it, as represented in the figure, the other times of their action, and their angular effects as half being hid below the ecliptic. Suppose this the times, those errors must diminish also on orbit folded down; it will coincide with the this account; and we can compute what those ecliptic in the circle N mcdn. Let EX repre errors will be for any diameter of the ring, and sent the axis of this orbit, perpendicular to its for any period of its revolution. We can tell, plane, and therefore inclined to the ecliptic. therefore, what would be the motion of the Since the moon gravitates to the sun in the di- nodes, the change of inclination, and deviation rection MS, which is all above the ecliptic, it is of the axis, of a ring which would touch the surplain that this gravitation has a tendency to face of the earth, and revolve in twenty-four draw the moon towards the ecliptic. Suppose hours; nay, we can tell what these motions this force to be such that it would draw the would be, should this ring adhere to the earth. moon down from M to i in the time that she They must be much less than if the ring were would have moved from M to t, in the tangent detached. For the disturbing forces of the ring to her orbit. By the combination of these mo must drag ng with it the whole globe of the tions the moon will desert her orbit, and describe carth. The quantity of motion which the disthe line M r, which makes the diagonal of the turbing forces would have produced in the ring 22 M м ort alone, will now, says Newton, be produced in principle, that the motion of the nodes of the the whole mass; and therefore the velocity must rigid ring is equal to the mean motion of the be as much less as the quantity of matter is nodes of the moon, has been most critically disgreater : but still all this can be computed. cussed by the first mathematicians, as a thing That there is such a ring on the earth is cer- which could neither be proved nor refuted. tain; for the earth is not a sphere, but an ellip- Frisius has at last shown it to be a mistake, and tical spheroid. Sir Isaac Newton, therefore, that the motion of the nodes of the ring is double made a computation of the effects of the disturb- the mean motion of the nodes of a single moon; ing force, and has exhibited a most beautiful ex- and that Newton's own principles should have ample of mathematical investigation. He first produced a precession of eighteen seconds and asserts that the earth must be an elliptical sphe- a quarter annually; which removes the difficulty roid, whose polar axis is to its equatorial diameter formerly mentioned. as 229 to 230, Then he demonstrates that if Sir Isaac Newton's third assumption, that the the sine of the inclination of the equator be called quantity of motion of the ring must be shared 7, and if t be the number of days (sidereal) in a with the included sphere, was acquiesced in by year, the annual motion of a detached ring will all his commentators, till D'Alembert and Euler, 3 71-7 in 1749, showed that it was not the quantity of te 360° X He then shows that motion round an axis of rotation which remained 4 t the same, but the quantity of momentum or rothe effect of the disturbing force on this ring is tatory effort. The quantity of motion is the to its effect on the matter of the same ring, dis- product of every particle by its velocity; that is, tributed in the form of an elliptical stratum (but by its distance from the axis; while its momenstill detached) as 5 to 2; therefore the motion tum, or power of producing rotation, is as the 31-72 square of that distance, and is to be had by of the nodes will be 360° X ; or 16' taking the sum of each particle multiplied by the 10 t square of its distance from the axis.' Since the 16" 24" annually. He then proceeds to show earth differs so little from a perfect sphere, this that the quantity of motion in the sphere is to makes no sensible difference in the result. It that in the equatorial ring revolving in the same will increase Newton's precession about threetime, as the matter in the sphere to the matter fourths of a second. in the ring, and as three times the square of a The source of Newton's mistake in the solution quadrantal arch to two squares of a diameter, of this intricate problem was first detected by jointly : then he shows that the quantity of mat- Mr. Landen, in the first volume of his Memoirs. ter in the terrestrial sphere is to that in the pro- That superior mathematician discovered that tuberant matter of the spheroid as 52900 to when a rigid annulus revolves with two motions, 461 (supposing all homogeneous). From these one in its own plane, and the other round one premises it follows that the motion of 16' 16" of its diameters, half the motive force acting 24"" must be diminished in the ratio of 10717 to upon the ring is counteracted by the centrifugal 100, which reduces it to 9' 07'' annually. And force arising from the compound motion, and this, he says, is the precession of the equinoxes, half only is efficacious or accelerating the plane occasioned by the action of the sun; and the of the annulus round its diameter. Mr. Landen rest of the 504', which is the observed preces- did not expressly demonstrate this, but it has sion, is owing to the action of the moon nearly been done very completely by Dr. Brinkley, in five times greater than that of the sun. This ap- the seventh volume of the Memoirs of the Irish peared a great difficulty ; for the phenomena of Academy. We cannot here pursue this subject; the tides show that it cannot much exceed twice but beg to refer the reader to Dr. Milner's paper the sun's force. in the Philosophical Transactions; to Dr. Abram The ingenuity of this process is justly celebrated Robertson's paper in the Philosophical Transby Daniel Bernouilli, who (in his Dissertation actions for 1807; to the Dissertation of Frisius on the Tides, which shared the prize of the French already specified ; and to the popular view of Academy with M'Laurin and Euler) says that this problem by M. Laplace in his Exposition, Newton saw through a veil what others could book'iv. ch. 13. hardly discover with a microscope in the light of To find the precession in right ascension and the meridian sun. His determination of the declination.- Put d= the declination of a star, form and dimensions of the earth, which is the and a = its right ascension; then their annual foundation of the whole process, is not offered as variations of precessious will be nearly as follow, any thing better than a probable guess, in re dif- viz. 20" •084 x cos. a = the annual precession ficillima; and it has been since demonstrated in declinat, and 46" .0619 + 20".084 X sin. a with geometrical rigor by M'Laurin. His next X tang.d = that of right ascension. PRECIÆ, precius, early, the twenty-first The main body of the sea being one, yet within order in Linnæus's fragments of a natural me divers precincts, hath divers names, so the catholick ibod; consisting of primrose, an early flowering church is in like sort divided into a number of dis Hooker. plant, and a few genera which agree with it in tinct societies. habit and structure. See BOTANY. This is the manner of God's dealing with those that have lived within the precincts of the church; they PRECINCT, n. s. Lat. præcinctus. Out- shall be condemned for the very want of true faith ward limit; boundary. and repentance. Perkins Through all restraint broke loose, he wings his tant is, falling or rushing headloug; hasty; hurway ried: the adverb corresponding : precipitate, to Nor far off heaven, in the precincts of light, urge or throw headlong; urge on with violence; Directly towards the new created world. Milton. hasten; hurry blindly; throw to the bottom by To find our hearthstone turned into a tomb, And round its once warm precincts palely lying a chemical process : as a verb neuter, to fall The ashes of our hopes, is a deep grief, headlong; fall to the bottom: precipitate as an Beyond a single gentleman's belief. Byron. adjective is synonymous with precipitant: as a noun substantive, it is a medical term for the PRECIOUS, adj. 2 Fr. precieur ; Latin red oxide of mercury : precipitately and precipiPRECIOUSLY, udv. pretiosus. Valuable; PRECIOUSNESS, n. s. Sof great worth ; costly; is, steep; headlong ; hasty ; rash. tation correspond with the adjective: precipitous often used in irony: the adverb and noun-substantive follow the senses of the adjective. Hadst thou been aught but goss’mer feathers, So many fathom down precipitating, A womman that hadde a boxe of alabastre of pre Thou’dst shiver like an egg. cious oynement cam to him and schedde out on the heed of him restynge. Wiclif. Matt. 26. Shakspeare. King Lear. Let them pile ten hills on the Tarpeian rock, The lips of knowledge are a precious jewel. Prov. xx. 15. That the precipitation might down-stretch Below the beam of sight, yet will I still Many things which are most precious, are neglected Be this to them. Id. Coriolanus. only because the value of them lieth hid. Hooker. Barcephas saith, it was necessary this paradise I never saw should be set at such a height, because the four Such precious deeds in one that promised nought rivers, had they not fallen so precipitate, could not But begg'ry and poor luck. Shakspeare. Cymbelino. have had sufficient force to thrust themselves under Its preciousness equalled the price of pearls. Wilkins. Raleigh. dark and unknown reasons, precipitated and banished That riches grow in hell; that soil may best the world into a nunnery. Bacon. Deserve the precious bane. Milton. As for having them obnoxious to ruin, if they be The index or forefinger was too naked whereto to commit their preciosities, and hath the tuition of the of fearful natures, it may do well; but, if they be thumb scarce unto the second joint., Browne. daring, it may precipitate their designs, and prove dangerous. Id. Barbarians seem to exceed them in the curiosity of their application of these preciosities. More. By strong water every metal will precipitate. Id. Fortune, conscious of your destiny, Separation is wrought by precipitation or sublimaEv'n then took care to lay you softly by ; tion; that is, a calling of the parts up or down, which is a kind of attraction. Id. And wrapp'd your fate among her precious things, The commotions in Ireland were so sudden and Kept fresh to be unfolded with your king's. Dryden. so violent, that it was hard to discern the rise, or More of the same kind, concerning these precious apply a remedy to that precipitant rebellion. King Charles. saints amongst the Turks, may be seen in Pietro della Valle. Loche. Monarchy, together with me, could not but be These virtues are the hidden beauties of a soul dashed in pieces by such a precipitous fall as they intended. Id. which make it lovely and precious in his sight, from Short intermittent and swift recurrent pains do whom no secrets are concealed. Addison's Spectator. precipitate patients into consumptions. Harvey. PRE’CIPICE, n. s. Fr. precipice; Lat. pre They were wont, upon a superstition, to precipicipitium. A headlong or perpendicular steep. tate a man from some high cliff into the sea, tying J ere long that precipice must tread, about him with strings many great fowls. Wilkins. Whence none return that leads unto the dead. Dear Erythræa, let not such blind fury Sandys. Precipitate your thoughts, nor set them working, You take a precipice for no leap of danger, Till time shall lend them better means And woo your own destruction. Shakspeare. Than lost complaints. Denham's Sophy. Where the water dasheth more against the bottom, The archbishop, too precipitate in pressing the rethere it moveth more swiftly and more in precipice ; ception of that which he thought a reformation, paid for in the breaking of the waves there is ever a pre- dearly for it. Clarendon. cipice. Bacon. Thither they haste with glad precipitance. Milton. Access, no horror turns away our eyes. Without longer pause, Denham. Downright into the world's first region throws Swift down the precipice of time it goes, His flight precipitant. Id. Paradise Lost. And sinks in minutes, which in ages rose. As the chymist, by catching at it too soon, lost Dryden. the philosophical elixir, so precipilancy of our underDrink as much as you can get; because a good standing is an occasion of error. Glanville, coachman never drives so well as when he is drunk; Though the attempts of some have been precipitous, and then shew your skill, by driving to an inch by a and their enquiries so audacious as to have lost precipice. Swift. themselves in attempts above humanity, yet have PRECIPITANCE, or ) Latin the enquiries of most defected by the way. PRECIP'ITANCY, n. s. precipi Browne's Vulgar Errours. PRECIP'ITANT, adj. The goddess guides her son, and turns him from tans. the light, PRECIPITANTLY, adv. Haste; Herself involved in clouds, precipitates her flight. PRECIPITATE, v.d., 0.9., adj. & n. s. rash Dryden. PRECIP'ITATELY, adv. haste; Thus framed for ill, he loosed our triple hold, PRECIPITATION, n. s. hurry: Advice unsafe, precipitous, and bold. Id. PRECIPITOUS. precipi How precious the time is, how precipitous the OC |