foundation of the whole process, is not offered as any Precession. thing better than a probable guess, in re difficillima; and it bas since been demonstrated with geometrical rigour by MLaurin. His next 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. Frisins has at least shown it to be a mistake, and that the motion of the nodes of the ring is double the mean inotion of the nodes of a single moon: and that Newton's own principles should have produced a precession of 18 seconds annually, which removes the difficulty formerly mentioned. Precession pose its motion of revolution to accelerate, or the time of a revolution to diminish; the linear effects of the disturbing forces being as the squares 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 ting which would touch the surface of the earth, and revolve in 24 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 greater: But still all this can be computed. 29 His deter Now there is such a ring on the earth for the earth is not a sphere, but an elliptical spheroid. Sir Isaac Newton therefore engaged in 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 inclina, tion of the equator be called #, and if t be the number of days (sidereal) in a year, the annual motion of 3V a detached ring will be 360°x: He then 4 t 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 of the nodes will be 360° X or 16′ 16′′ 24′′ annually. He then proceeds to show, that the quantity of motion in the sphere is to that in an 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 quadrantral 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 52900 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" 7"" annually. And this (he says) is the precession of the equinoxes, occasioned by the action of the sun; and the rest of the 50" 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. Nothing can exceed the ingenuity of this process. sination Justly does his celebrated and candid commentator, Daof the form niel Bernoulli, say (in his Dissertation on the Tides, and dimen- which shared the prize of the French Academy with M'Laurin and Euler,), that Newton saw through a veil what others could hardly discover with a microscope by MLau- in the light of the meridian sun. His determination of the form and dimensions of the earth, which is the sions of the earth de. Inonstrated rin. His 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 product of every particle by its velocity; that is, by 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 three-fourths of a second. 30 We proceed now to the examination of this pheno- Examinamenon upon the fundamental principles of mechanics. tion of the Because the mutual gravitation of the particles of phenomematter in the solar system is in the inverse ratio of the non of presquares of the distance, it follows, that the gravitations mechanical of the different parts of the earth to the sun or to the principles. moon are unequal. The nearer particles gravitate more than those that are more remote. cession on Let PQpE (fig. 3.) be a meridional section of the Fig. 3 terrestrial sphere, and POpq the section of the inscribed sphere. Let CS be a line in the plane of the ecliptic passing through the sun, so that the angle ECS is the sun's declination. Let NCM be a plane passing through the centre of the earth at right angles to the plane of the meridian PQPE; NCM will therefore be the plane of illumination. In consequence of the unequal gravitation of the matter of the earth to the sun, every particle, such as B, is acted on by a disturbing force parallel to CS, and proportional to BD, the distance of the particle from the plane of illumination; and this force is to the gravitation of the central particle to the sun, as three times BD to CS, the distance of the earth from the sun. Let A Ba be a plane passing through the particle B, parallel to the plane EQ of the equator. This section of the earth will be a circle, of which Aa is a diameter, and Qq will be the diameter of its section with the inscri bed sphere. These will be two concentric circles, and the ring by which the section of the spheroid exceeds the section of the sphere, will have AQ for its breadth; Pp is the axis of figure. Let EC be represented by the symbol a b П m 2 of an arch, and the sum of its square and the square of Precession. its corresponding cosine is equal to the square of the radius. Therefore the sum of all the squares of the sines, together with the sum of all the squares of the f cosines, is equal to the sum of the same number of squares of the radius; and the sum of the squares of the sines is equal to the sum of the squares of the corresponding cosines: therefore the sum of the squares of the radius is double of either sum. Therefore.QL .BL¦n QL.QL'. In like manner the sum of the number П.QL of CL's will be п.QL.CL'. These sums, taken for the semicircle, are п.QL.QL3, and n It is evident, that with respect to the inscribed sphere, the disturbing forces are completely compensated, for every particle has a corresponding particle in the adjoining quadrant, which is acted on by an equal and opposite force. But this is not the case with the protuberant matter which makes up the spheroid. The seg-n.QL.CL', or п.QL.¿QL3, and п.QL.÷CL2: there ments NS sn and MT tm are more acted on than the segments NT tn and MS sm; and thus there is produced a tendency to a conversion of the whole earth, round an axis passing through the centre C, perpendicular to the plane PQP E. We shall distinguish this motion from all others to which the spheroid may be subject, by the name LIBRATION. The axis of this libration is always perpendicular to that diameter of the equator over which the sun is, or to that meridian in which he is. PROB I. To determine the momentum of libration corresponding to any position of the earth respecting the sun, that is, to determine the accumulated energy of the disturbing forces on all the protuberant matter of the spheroid. Let B and b be two particles in the ring formed by the revolution of AQ, and so situated that they are at equal distances from the plane NM; but on opposite sides of it. Draw BD, bd, perpendicular to NM, and FLG perpendicular to LT. Then, because the momentum, or power of producing rotation, is as the force and as the distance of its line of direction from the axis of rotation, jointly, the combined momentum of the particles B and b will be f.BD.DC-f.bd.dc, (for the particles B and bare urged in contrary directions). But the momentum of Bis f.BF.DC+f.FD.DC, and that of b is f.b G. dC — f.dG.dC; and the combined momentum is f.BF.Ddƒ.FD.DC+dC, = 2ƒ.BF.LF—2f.LT.TC. Because m and n are the sine and cosine of the angle ECS or LCT, we have LT=m.CL, and CT=n.CL, and LF m.BL, and BF=n.BL. This gives the momentum 2 fm n BL3-CL3. The breadth AQ of the protuberant ring being very small, we may suppose, without any sensible error, that all the matter of the line AQ is collected in the point Q; and, in like manner, that the matter of the whole ring is collected in the circumference of its inner circle, and that B and 6 now represent, not single particles, but the collected matter of lines such as AQ, which terminate at B and b. The combined momentum of two such lines will therefore be 2 m n f.AQ.BL-CL3. Let the circumference of each parallel of latitude be divided into a great number of indefinitely small and equal parts. The number of such parts in the circumference of which Q q is the diameter, will be ПQL. To each pair of these there belongs a momentum 2mnf .AQ BL-CL'. The sum of all the 'quares of BL, which can be taken round the circle, is one half of as many squares of the radius CL: for BL is the sine fore the momentum of the whole ring will be 2 m n f .AQ.QL.n(QL-CL): for the momentum of the ring is the combined momentum of a number of pairs, and this number is in.QL. d b By the ellipse we have OC: QL=EO: AQ, and EO d AQ=QL OC=QL; therefore the momentum of the rings is 2 m n ƒ— QL3n (¿QL'—'CL'), =m nf QL3n (¿QL2-CL1): but QL'=ba—x2; therefore QL*—CL1={b'—{x2—x3‚ ={V—{x2, = ; therefore the momentum of the ring is m nƒ ̃-¤(b2--s') 2 This formula does not express any motion, but only a pressure tending to produce motion, and particularly tending to produce a libration by its action on the cohering matter of the earth, which is affected as a number of levers. It is similar to the common mechanical formula w.d, where w means a weight, and d its distance from the fulcrum of the lever. It is worthy of remark, that the momentum of this protuberant matter is just one fifth of what it would be if it were all collected at the point O of the equator: for the matter in the spheroid is to that in the inscribed sphere as a to 2, and the contents of the inscribed sphere is 3. Therefore a ; a2—ba—‡ñ 3 : ‡пb® which is the quantity of protuberant matter. We αλ Precession. We may, without sensible error, suppose 31 32 When a rigid body is made to turn round any axis Precession. by the action of an external force, the quantity of momentum produced (that is, the sum of the products of every particle by its velocity and by its distance from the axis) is equal to the momentum or similar product of the moving force or forces. If an oblate spheroid, whose equatorial diameter is a and polar diameter b, be made to librate round an equatorial diameter, and the velocity of that point of the equator which is farthest from the axis of libration be v, the momentum of the spheroid is nabv. 15 The two last are to be found in every elementary book of mechanics. 33 Also, because the sum of all the rectangles OH.HC Let AN an (fig. 4.) be the plane of the earth's equa- Fig. 4 20 Thus far Sir Isaac Newton proceeded with mathema- of moons is the same with the mean motion of the nodes Effects of Prob. 2. To determine the deviation of the axis, and If a rigid body is turning round an axis A, passing the earth by a meridian passing through the sun, so distance I from the axis, or it is the space which would 73 Letv be the velocity produced in the point A, the v fdi b or very nearly mn ƒ di, because I very nearly. Also, Let be the momentary angle of diurnal rotation. Precession. OP, perpendicular to the plane of the equator ≈ 0 %, and and down like the arm of a balance. On this account Precession therefore situated in the plane ZP ≈; and it turns round this motion is very properly termed libration; but this very slow libration, compounded with the incomparably this axis with the angular velocity – . It has received swifter motion of diurnal rotation, produces a third motion extremely different from both. At first the north Fig 4. an impulse, by which alone it would librate round the pole of the earth inclines forward toward the sun; after mnfdi a axis Zx, with the angular velocity It will therefore turn round neither axis (N° 31.), but round a third axis OP', passing through O, and lying in the plane ZP %, in which the other two are situated, and the sine P' of its inclination to the axis of libration Z z will be to the sine Pp of its inclination to the axis mnf di OP of rotation as t to a Now A, in fig. 4. is the summit of the equator both, of libration and rotation; mnf dia is the space described by its libration in the time i; and ar is the space or arch Ar (fig. 4.) described in the same time by its rotation therefore, taking Ar to A c (perpendicular to the plane of the equator of rotation, and lying in the equator of libration,) as ar to m nf di, and completing the parallelogram Armc, Am will be the compound motion of A (N° 31.), and ar:mnf di mnfdi which will be the tangent of the angle ed by the composition of rotations. In consequence of this change of position, the plane of the equator no longer cuts the plane of the ecliptic in the line N n. The plane of the new equator cuts the former equator in the line AO, and the part AN of the former equator lies between the ecliptic and the new equator AN', while the part A n of the former equator is above the new one A n'; therefore the new node N', from which the point A was moving, is removed to the westward, or farther from A; and the new node n', to which A is approaching, is also moved westward, or nearer to A; and this happens in every position of A. The nodes, therefore, or equinoctial points, continually shift to the westward, or in a contrary direction to the rotation of the earth; and the axis of rotation always deviates to the east side of the meridian which passes through the sun. This account of the motions is extremely different from what a person should naturally expect. If the earth were placed in the summer solstice, with respect to us who inhabit its northern hemisphere, and had no rotation round its axis, the equator would begin to approach the ecliptic, and the axis would become more upright; and this would go on with a motion continually accelerating, till the equator coincided with the ecliptic. It would not stop here, but go as far on the other side, till its motion were extinguished by the opposing forces; and it would return to its former position, and again begin to approach the ecliptic, playing up a long course of years it will incline to the left band, as viewed from the sun, and be much more inclined to the ecliptic, and the plane of the equator will pass through the sun. Then the south pole will come into view, and the north pole will begin to decline from the sun; and this will go on (the inclination of the equator diminishing all the while) till, after a course of years, the north pole will be turned quite away from the sun, and the inclination of the equator will be restored to its original quantity. After this the phenomena will have another period similar to the former, but the axis will now deviate to the right hand. And thus, although both the earth and sun should not move from their places, the inhabitants of the earth would have a complete succession of the seasons accomplished in a period of many centuries. This would be prettily illustrated by an iron ring poised very nicely on a cap like the card inciding with the point of the cap, so that it may whirl of a mariner's compass, having its centre of gravity coround in any position. As this is extremely difficult which will cause the ring to maintain a horizontal posito execute, the cap may be pierced a little deeper, ing very steadily, and pretty briskly, in the direction tion with a very small force. When the ring is whirl of the hours of a watch-dial, hold a strong magnet above the middle of the nearer semicircle (above the 6 hour point) at the distance of three or four inches. We shall immediately observe the ring rise from the 9 hour point, and sink at the 3 hour point, and gradually acquire a motion of precession and nutation, such as has been described. If the earth be now put in motion round the sun, or the sun round the earth, motions of libration and deviation will still obtain, and the succession of their different phases, if we may so call them, will be perfectly analogous to the above statement. But the quantity of deviation, and change of inclination, will now be prodigiously diminished, because the rapid change of the sun's position quickly diminishes the disturbing forces, annihilates them by bringing the sun into the plane of the equator and brings opposite forces into action. We see in general that the deviation of the axis is always at right angles to the plane passing through the sun, and that the axis, instead of being raised from the ecliptic, or brought nearer to it, as the libration would occasion, deviates sidewise; and the equator, instead of being raised or depressed round its east and west points, is twisted sidewise round the north and south points; or at least things have this appearance; but we must now attend to this circumstance more minutely. The composition of rotation shows us that this change of the axis of diurnal rotation is by no means a translation of the former axis (which we may suppose to be the axis of figure) into a new position, in which it again becomes the axis of diurnal motion; nor does the equator of figure, that is, the most prominent section of the terrestrial spheroid, change its position, and in this new position continue to be the equator of rotation. This was indeed supposed by Sir Isaac New ton; what nutation will be accumulated after any given time Precession. of action. For this purpose we must ascertain the precise deviation which the disturbing forces are competent to produce. This we can do by comparing the momentum of libration with the gravitation of the earth to the sun, and this with the force which would retain a body on the equator while the earth turns round its axis. Precession. ton; and this supposition naturally resulted from the train of reasoning which he adopted. It was strictly true of a single moon, or of the imaginary orbit attached to it; and therefore Newton supposed that the whole carth did in this manner deviate from its former position, still, however, turning round its axis of figure. In this he has been followed by Walmesly, Simpson, and most of his commentators. D'Alembert was the first who entertained any suspicion that this might not be certain; and both he and Euler at last showed that the new axis of rotation was really a new line in the body of the earth, and that its axis and equator of figure did not remain the axis and equator of rotation. They ascertained the position of the real axis by means of a most intricate analysis, which obscured the connection of the different positions of the axis with each other, and gave us only a kind of momentary information. Father Frisius turned his thoughts to this problem, and fortunately discovered the composition of rotations as a general principle of mechanical philosophy. Few things of this kind have escaped the penetrating eye of Sir Isaac Newton. Even this principle had been glanced at by him. He affirms it in express terms with respect to a body that is perfectly spherical (cor. 22. prop. 66. B. I.). But it was reserved for Frisius to demonstrate it to be true of bodies of any figure, and thus to enrich mechanical science with a principle which gives simple and elegant solutions of the most difficult problems. Fig. 6. But here a very formidable objection naturally offers itself. If the axis of the diurnal motion of the heavens is not the axis of the earth's spheroidal figure, but an imaginary line in it, round which even the axis of figure must revolve; and if this axis of diurnal rotation has so greatly changed its position, that it now points at a star at least 12 degrees distant from the pole observed by Timochares, how comes it that the equator has the very same situation on the surface of the earth that it had in ancient times? No sensible change has been observed in the latitude of places. The answer is very simple and satisfactory: Suppose that in 12 hours the axis of rotation has changed from the position PR (fig. 6.) to pr, so that the north pole, The gravitation of the earth to the sun is in the pro A4 of the angle. This being extremely small, the sine may angle. Now, substitute for it the value now found, viz. T and we obtain an angle of deviation w= ts d can appear. But it is convenient, for other reasons, to therefore wrx 3 13 m n a3—b1 2 a3 and this is the form a2 instead of being at P, which we may suppose to be a particular mountain, is now at p. In this 12 hours the 3. mountain P, by its rotation round pr, has acquired the position. At the end of the next 12 hours, the axis of rotation has got the position, and the axis of figuren, and this is the simplest form in which it has got the position pr, and the mountain P is now at p. Thus, on the noon of the following day, the axis of figure PR is in the situation which the real axis of rotation occupied at the intervening midnight. This goes on continually, and the axis of figure follows the position of the axis of rotation, and is never further removed from it than the deviation of 12 hours, which does not exceed th part of one second, a quantity altogether imperceptible. Therefore the axis of figure will always sensibly coincide with the axis of rotation, and no change can be produced in the latitudes of places on the surface of the earth. Application We have hitherto considered this problem in the most of this rea- general manner; let us now apply the knowledge we soning to have gotten of the deviation of the axis or of the moand preces-mentary action of the disturbing force to the explanation of the phenomena: that is, let us see what precession and 34 nutation sion. in which we shall now employ it.. 2T2 The small angler 31 mn is the angle in which Let |