The most diurnal revolution of all the celestial motions is the diurnal revolution of the starry heavens. The whole appears to the starry turn round an imaginary axis, which passes through two opposite points of the heavens, called the poles. One of these is in our sight, being very near the star α in the tail of the Little Bear. The great circle which is equidistant from both poles divides the heavens into the northern and southern hemispheres, which are equal. It is called the equator, and it cuts the horizon in the east and west points, and every star in it is 12 sidereal hours above and as many below the horizon, in each revolution.

The sun's motions determine the length of day Observations, and night, and the vicissitudes of the seasons. By actions of the long series of observations, the shepherds of Asia were able to mark out the sun's path in the heavens; he being always in the opposite point to that which comes to the meridian at midnight, with equal but opposite declination. Thus they could tell the stars among which the sun then was, although they could not see them. They discovered that his path was a great circle of the heavens, afterwards called the ECLIPSE; which cuts the equator in two opposite points, dividing it, and being divided by it, into two equal parts. They farther observed, that when the sun was in either of these points of intersection, his circle of diurnal revolution coincided with the equator, and therefore the days and nights were equal. Hence the equator came to be called the EQUINOCTIAL LINE, and the points in which it cuts the ecliptic were called the EQUINOCTIAL POINTS, and the sun was then said to be in the equinoxes. One of these was called the VERNAL and the other the AUTUMNAL EQUINOX.

It was evidently an important problem in practical astronomy to determine the exact moment of the sun's time of occupying these stations; for it was natural to compute sun's occupation of the year from that moment. Accordingly, this has been the leading problem in the astronomy of equinoctial all nations. It is susceptible of considerable precision, without any apparatus of instruments. It is only necessary to observe the sun's declination on the noon of two or three days before and after the equinoctial day. On two consecutive days of this number, his declination must have changed from north to south, or from south to north. If his declination on one day was observed to be 21' north, and on the next 5' south, it follows that his declination was nothing, or that he was in the equinoctial point about 23' after seven in the morning of the second day. Knowing the precise moments, and knowing the rate of the sun's motion in the ecliptic, it is easy to ascertain the precise point of the ecliptic in which the equator intersected it.

By a series of such observations made at Alexandria Hipparchus between the years 161 and 127 before Christ, Hipparchus's discoveries. Thus the father of our astronomy found that the point of the autumnal equinox was about six degrees to the eastward of the star called Spica Virginis. Eager to determine every thing by multiplied observations, he ransacked all the Chaldean, Egyptian, and other records, to which his travels could procure him access, for observations of the same kind; but he does not mention his having found any. He found, however, some observations of Aratus and Timocharis, made about 150 years before. From these it appeared evident that the point of the autumnal equinox was then about eight degrees east of the same star. He discusses these observations with great sagacity and rigour; and, on their authority, he affirms that the equinoctial points are not fixed in the heavens, but move to the westward about a degree in 75 years or somewhat less.

This motion is called the Precession of the Equinoxes, because by it the time and place of the sun's equinoctial station precedes the usual calculations: it is fully confirmed by all subsequent observations. In 1750 the autumnal equinox was observed to be 26° 21' westward of Spica Virginis. Supposing the motion to have been uniform during this period of ages, it follows that the annual precession is about 50" 3"; that is, if the celestial equator cuts the ecliptic in a particular point on any day of this year, it will on the same day of the following year cut it in a point 50" 3" to the west of it, and the sun will come to the equinox 20' 23" before he has completed his round of the heavens. Thus the equinoctial or tropical year, or true year of seasons, is so much shorter than the revolution of the sun or the sidereal year.

It is this discovery that has chiefly immortalized the name of Hipparchus, though it must be acknowledged that all his astronomical researches have been conducted with the same sagacity and intelligence. It was natural therefore for him to value himself highly for the discovery; for it must be admitted to be one of the most singular that has been made, that the revolution of the whole heavens should not be stable, but its axis continually changing. For it must be observed, that since the equator changes its position, and the equator is only an imaginary circle, equidistant from the two poles or extremities of the axis; these poles and this axis must equally change their positions. The equinoctial points make a complete revolution in about 25,745 years, the equator being all the while inclined to the ecliptic in nearly the same angle. Therefore the poles of this diurnal revolution must describe a circle round the poles of the ecliptic at the distance of about 23° 5 degrees in 25,745 years; and in the time of Timocharis, the north pole of the heavens must have been 30 degrees eastward of the place where it now is.

Hipparchus has been accused of plagiarism and insincerity in this matter. It is now very certain that the precession of the equinoxes was known to the astronomers of India many ages before the time of Hipparchus. It appears also that the Chaldeans had a pretty accurate knowledge of the year of seasons. From their faros we deduce their measure of this year to be 365 days 5 hours 49 minutes and 11 seconds, exceeding the truth only by 26", and much more exact than the year of Hipparchus. They had also a sidereal year of 365 days 6 hours 11 minutes. Now what could occasion an attention to two years, if they did not suppose the equinoxes moveable? The Egyptians also had a knowledge of something equivalent to this: for they had discovered that the dog-star was no longer the faithful forewarning of the overflowing of the Nile; and they pins for the combined him with the star Fomalhaut* in their mythological calendar. This knowledge is also involved in the Egyptian precepts of the Chinese astronomy, of much older date than the time of Hipparchus.

But all these acknowledged facts are not sufficient for depriving Hipparchus of the honour of the discovery. But falsely, very, or fixing on him the charge of plagiarism. This motion was a thing unknown to the astronomers of the Alexandrian school, and it was pointed out to them by Hipparchus in the way in which he ascertained every other position in astronomy, namely, as the mathematical result of actual observations, and not as a thing deducible from any opinions on other subjects related to it. We see him on all other occasions, eager to confirm his own observations, and his deductions from them, by everything he could pick up from other astronomers; and he even adduced the above-mentioned practice of the Egyptians in corroboration of his doctrine. It is more than probable then that he did not know anything more. Had he known the Indian precession of 54" annually, he had no temptation whatever to withhold him from using it in preference to one which he acknowledges to be inaccurate, because deduced from the very short period of 150 years, and from the observations of Timocharis, in which he had no great confidence.

This motion of the starry heavens was long a matter of discussion, as a thing for which no physical reason motions could be assigned. But the establishment of the Copernican system reduced it to a very simple affair; the Copernican motion which was thought to affect all the heavenly bodies, is now acknowledged to be a deception, or a false judgment from the appearances. The earth turns round its own axis while it revolves round the sun, in the same manner as we may cause a child's top to spin on the brim of a millstone, while the stone is turning slowly round its axis. If the top spin steadily, without any wavering, its axis will always point to the zenith of the heavens; but we frequently see, that while it spins briskly round its axis, the axis itself has a slow conical motion round the vertical line, so that, if produced, it would slowly describe a circle in the heavens round the zenith point. The flat surface of the top may represent the terrestrial equator, gradually turning itself round on all sides. If this top were formed like a ball, with an equatorial circle on it, it would represent the whole motion very prettily, the only difference being, that the spinning motion and this wavering motion are in the same direction; whereas the diurnal rotation and the motion of the equinoctial points are in contrary directions. Even this dissimilarity may be removed, by making the top turn on a cap, like the card of a mariner's compass.

It is now a matter fully established, that while the earth revolves round the sun from west to east, in the plane of the ecliptic in the course of a year, it turns round its own axis from west to east in 23h 56' 4", which axis is inclined to this plane in an angle of nearly 23° 28'; and that this axis turns round a line perpendicular to the ecliptic in 25,745 years from east to west, keeping nearly the same inclination to the ecliptic. By this means, its pole in the sphere of the starry heavens describes a circle round the pole of the ecliptic at the distance of $23^\circ 28'$ nearly. The consequence of this must be, that the terrestrial equator, when produced to the sphere of the starry heavens, will cut the ecliptic in two opposite points, through which the sun must pass when he makes the day and night equal; and that these points must shift to the westward, at the rate of $50\frac{1}{7}$ seconds annually, which is the precession of the equinoxes. Accordingly this has been the received doctrine among astronomers for nearly three centuries, and it was thought perfectly conformable to appearances.

But Dr Bradley, the most sagacious of modern astronomers, hoped to discover the parallax of the earth's orbit by observations of the actual position of the pole of the celestial revolution. Dr Hooke had attempted this before, but with very imperfect instruments. The art of observing being now prodigiously improved, Dr Bradley resumed this investigation. It will easily appear, that if the earth's axis keeps parallel to itself, its extremity must describe in the sphere of the starry heavens a figure equal and parallel to its orbit round the sun; and if the stars be so near that this figure is a visible object, the pole of diurnal revolution will be in different distinguishable points of this figure. Consequently, if the axis describes the cone already mentioned, the pole will not describe a circle round the pole of the ecliptic, but will have a looped motion along this circumference, similar to the absolute motion of one of Jupiter's satellites, describing an epicycle whose centre describes the circle round the pole of the ecliptic.

He accordingly observed such an epicyclical motion, and thought that he had now overcome the only difficulty in the Copernican system; but, on maturely considering his observations, he found this epicycle to be quite inconsistent with the consequences of the annual parallax, and it puzzled him exceedingly. One day, while taking the amusement of sailing about 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 immediately suggested to him the cause 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 unwittingly established an incontrovertible argument for the truth of the Copernican system, and immortalized his name by his discovery of the Aberration of the stars.

He now engaged in a series of observations for ascertaining all the circumstances of this discovery. In the course of these, which were continued for 28 years, he discovered another epicyclical motion of the pole of the heavens, which was equally curious and unexpected. 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. He also observed, that the period of this epicyclical motion was 18 years and seven months. It struck him, that this was precisely the period of the revolution of the nodes of the moon's orbit. He gave a brief account of these results to Lord Macclesfield, then president of the Royal Society, in 1747. 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 SPQ a circle distant from it $23^\circ 28'$, representing the circle described by the pole of the equator during one revolution of the equinoctial points. Let P be the place of this last mentioned pole at some given time. Round P describe a circle ABCD, whole diameter AC is $18''$. The real situation of the pole will be in the circumference of this circle; and its place, in this circumference depends on the place of the moon's ascending or descending node. Draw EPF and GPL perpendicular to it; let poled to the G.L be the colure of the equinoxes, and EF the colure of the 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 pole is in B. When the node is in the autumnal equinox at G, the pole is at C; and when the node 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 ABCD, three signs or $92''$ more advanced.

Dr Bradley, by comparing together a great number of observations, 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 ABCD, making the longer axis AC circle, $18''$, and the shorter, BD, $16''$. Mr d'Alembert determined, by the physical theory of gravitation, the axes to be $18''$ and $13''$.

These observations, and this mathematical theory, must be considered as so many facts in astronomy, and we must deduce from them the methods of computing the places of all celestial phenomena, agreeable to the universal practice of determining every point of the heavens by its longitude, latitude, right ascension, and declination.

It is evident, in the first place, that 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 measured by the arch of a great circle intercepted between their poles. Now, if the pole be in O instead of P, it is plain that the obliquity is measured by EO 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 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 shewed, 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 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 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.

The quantity of this change of obliquity is easily ascertained. It is evident, from what has been already said, that when the pole is in O, the arch ADCO is equal to the node's longitude from the vernal equinox, and and that PM is its cosine; and (on account of the smallness of AP in comparison of EP) PM may be taken for the change of the obliquity of the ecliptic. This is therefore \( g'' \times \text{cof. long. node} \), and is additive to the mean obliquity, while O is in the semicircle BAD, that is, while the longitude of the node is from 9 signs to 3 signs; but subtractive while the longitude of the node changes from 3 to 9 signs.

But the nutation also affects 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. It was this circumstance which made it necessary for us to consider it in this place, while expressly treating of this precession. Let us attend to this derangement 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 P or in O, EP or EO is the solstitial colure. Let SE be any fixed star or planet, and let SE be a meridian or circle of longitude; draw the circles of declination PS, OS, and the circles M'EM'', mEm', perpendicular to PE, OE.

If the pole were in its mean place P, the equinoctial 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 mEm' will pass through the equinoctial points. The equinoctial points must therefore be to the westward 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 nutation of the earth's axis, is the angle OEP, or \( \frac{OM}{OE} \).

OM is the sine of the angle CPO, which, by what has been already observed, is equal to the longitude of the node: Therefore OM is equal to \( g'' \times \text{long. node} \), and \( \frac{OM}{OE} \) is equal to \( \frac{g'' \times \text{fin. long. node}}{\text{fin. oblit. elip.}} \). This equation is additive to the mean longitude of the star when O is in the semicircle CBA, 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 ADC, 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 five signs, and additive to it while the node is in the last five signs.

This equation of longitude is the same for all the stars, for their longitude is reckoned on the ecliptic (which is here supposed invariable); and therefore is affected only by the variation of the point from which the longitude is computed.

The right ascension, being computed on the equator, suffers a double change. It 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 Precession S from the solstitial colure is measured by the angle SPE, 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 SOE, and the difference of SPE and SOE is the equation of right ascension. The angle SOE consists of two parts, GOE and GOS; GOE remains the same wherever the star S is placed, but GOS varies with the place of the star.

We must first find the variation by which GPE becomes GOE, which variation is common to all the stars. The triangles GPE, GOE, have a constant side GE, and a constant angle G; the variation PO of the side GP is extremely small, and therefore the variation of the angles may be computed by Mr Cotes's Fluxionary Theorems. See Simpson's Fluxions, § 253, &c. As the tangent of the side EP, opposite to the constant angle G, is to the sine of the angle EPG, opposite to the constant side EG, so is PO the variation of the side GP, adjacent to the constant angle, to the variation \( x \) of the angle GPO, opposite to the constant side EG. This gives \( x = \frac{g'' \times \text{fin. long. node}}{\text{tang. obl. elip.}} \). This is subtractive from the mean right ascension for the first five signs of the node's longitude, and additive for the last five signs. This equation is common to all the stars.

The variation of the other part SOG of the angle, other which depends on the different position of the hour circles, PS and OS, which causes them to cut the equation in different points, where the arches of right ascension terminate, may be discovered as follows: The triangles SPG, SOG, have a constant side SG, and a constant angle G. Therefore, by the same Cotean theorem, tan. SP : fin. SPG = PO : y, and y, or the second part of the nutation in right ascension, \( g'' \times \text{fin. diff. R. A. of star and node} \).

\[ \tan(\text{declin. star}) = \frac{g'' \times \text{fin. diff. R. A. of star and node}}{\text{cotan. declin. star}} \]

The nutation also affects the declination of the stars: Nutation For SP, the mean ecodeclination, is changed into SO. affects the Suppose a circle described round S, with the distance of the stars. SO cutting SP in f; then it is evident that the equation of declin. is \( Pf = PO \times \text{cof. OP} = g'' \times \text{sign r. ascen. of star—long. of node} \).

Such are the calculations in constant use in our astro-nomical recourses, founded on Machin's Theory. When exact mode still greater accuracy is required, the elliptical theory of calculation 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 OM in the ratio of 18 to 13.4, and a corresponding diminution of the angle CPO. The detail of it may be seen in De la Lande's Astronomy, art. 2874; but is rather foreign to our present purpose of explaining the precession of the equinoxes. 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 all these equations, both on the circular and elliptical hypothesis. And still more to abridge calculations, which occur in reducing every astronomical observation, when the place of a phenomenon is deduced from a comparison with known stars, there have been published tables of nutation and preces- Precession, for some hundreds of the principal stars, for every position of the moon's node and of the sun.

It now remains to consider the precession of the equinoctial points, with its equations, arising from the nutation of the earth's axis as a physical phenomenon, and to endeavour to account for it upon those mechanical principles which have so happily explained all the other phenomena of the celestial motions.

This did not escape the penetrating eye of Sir Isaac Newton; and he quickly found it to be a consequence, and the most beautiful proof, of the universal gravitation of all matter to all matter; and there is no part of his immortal work where his sagacity and fertility of resource shine more conspicuously than in this investigation. It must be acknowledged, however, that Newton's investigation is only a firewood 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 pick out in such a complication of arbitrary 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 all mechanical problems. The most elaborate and accurate dissertations on the precession of the equinoxes are those of Sylvabella and Walmsley, in the Philosophical Transactions, published about the year 1754; that of Thomas Simpson, published in his Miscellaneous Tracts; that of Father Frisius, in the Memoirs of the Berlin Academy, and afterwards with great improvements, 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 prize in the Academy of Paris in 1769. We think the dissertation of Father Frisius the most perspicuous of them all, being conducted in the method of geometrical analysis; whereas most of the others proceed in the fluxionary and symbolic method, which is frequently deficient in distinct notions of the quantities under consideration, and therefore does not give us the same perspicuous conviction of the truth of the results. In a work like ours, it is impossible to do justice to the problem, without entering into a detail which would be thought extremely disproportionate to the subject by the generality of our readers. Yet those who have the necessary preparation of mathematical knowledge, and wish to understand the subject fully, will find enough here to give them a very distinct notion of it; and in the article Rotation, they will find the fundamental theorems, which will enable them to carry on the investigation. We shall first give a short sketch of Newton's investigation, which is of the most palpable and popular kind, and is highly valuable, not only for its ingenuity, but also because it will give our unlearned readers distinct and satisfactory conceptions of the chief circumstances of the whole phenomena.

Let S (fig. 2.) be the sun, E the earth, and M the moon, moving in the orbit NMCD n, which cuts the plane of the ecliptic in the line of the nodes N n, 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 c d n. Let EX represent the axis of this orbit, perpendicular to its plane, and therefore inclined to the ecliptic. Since the moon gravitates to the sun in the direction MS, which is all above Precession, 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 it 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 M & n', lying between the orbit MCD 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 EX will no longer be perpendicular to it; but there will be another line E x, 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 nodes of the orbit in this new position is N' n'. Also the angle MN' m is less than the angle MN m.

Thus the nodes shift their places in a direction opposite to that of her motion, or move to the westward; 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 eastward. But, in general, the inclination increases from the time that the nodes are in the line of syzygy, till they get into quadrature, after which it diminishes till the nodes are again in syzygy. The nodes advance only while they are in the octants after the quadratures, and while the moon passes from 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; therefore the nodes of this rigid ring will continually recede, except when the plane of the ring passes through the sun, that is, when the nodes are in syzygy; and (says Newton) the motion of these nodes will be the same with the mean motion of the nodes of the orbit of one moon. The inclination of this ring to the ecliptic will be equal to the mean inclination of the moon's orbit during any one revolution which has the same situation of the nodes. It will therefore be least of all when the nodes are in quadrature, and will increase till they are in syzygy, and then diminish till they are again in quadrature.

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. Sup- foundation of the whole process, is not offered as any precession better than a probable guess, in re difficillima; and it has since been demonstrated with geometrical rigour by M'Laurin.

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. Frisius has at least 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 of 18½ seconds annually, which removes the difficulty formerly mentioned.

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 rotary 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.

We proceed now to the examination of this phenomenon upon the fundamental principles of mechanics. Because the mutual gravitation of the particles of the matter in the solar system is in the inverse ratio of the squares of the distance, it follows, that the gravitations mechanical of the different parts of the earth to the sun or to the moon are unequal. The nearer particles gravitate more than those that are more remote.

Let PQpE (fig. 3.) be a meridional section of the terrestrial sphere, and POPq the section of the inclosed 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 ABa 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 inclosed sphere. There 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

\[ \frac{OC}{PC} = \frac{a^2 - b^2}{a + b} \]

L12 CL 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 segments NS and MT are more acted on than the segments NT and MS; 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 PQ and 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 b are urged in contrary directions). But the momentum of B is f.BF, DC + f.FD, DC, and that of b is f.bG, DC - f.dG, DC; and the combined momentum is f.BF, DD - f.FD, DC + dC, = 2f.BF, LF - 2f.LT, TC.

Because m and n are the fine and cofine 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 = 2m.nf.BL^2 - CL^2.

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 b 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 2m.nf.AQ.BL^2 - CL^2.

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 Qq is the diameter, will be Pi.QL. To each pair of these there belongs a momentum 2m.nf.AQ.BL^2 - CL^2. The sum of all the squares of BL, which can be taken round the circle, is one half of as many squares of the radius CL: for BL is the fine of an arch, and the sum of its square and the square of its corresponding cofine is equal to the square of the radius. Therefore the sum of all the squares of the fines, together with the sum of all the squares of the cofines, is equal to the sum of the same number of squares of the radius; and the sum of the squares of the fines is equal to the sum of the squares of the corresponding cofines: therefore the sum of the squares of the radius is double of either sum. Therefore \( \int_{\text{Pi}} \text{QL} \cdot \text{BL}^2 = \frac{1}{2} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \).

In like manner the sum of the number Pi.QL of CL^2s will be \( \frac{1}{2} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \). These sums, taken for the semicircle, are \( \frac{1}{4} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \), and \( \frac{1}{4} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \); or \( \frac{1}{4} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \), and \( \frac{1}{4} \text{Pi} \cdot \text{QL} \cdot \text{CL}^2 \): therefore the momentum of the whole ring will be \( 2m.nf.AQ \cdot \text{QL} \cdot \text{Pi} \cdot \left( \frac{1}{4} \text{QL}^2 - \frac{1}{4} \text{CL}^2 \right) \): for the momentum of the ring is the combined momentum of a number of pairs, and this number is \( \frac{1}{2} \text{Pi} \cdot \text{QL} \).

By the ellipse we have OC : QL = EO : AQ, and AQ = QL.OC = QL.d/b; therefore the momentum of the rings is \( 2m.nf.d^2 \cdot \text{QL} \cdot \text{Pi} \cdot \left( \frac{1}{4} \text{QL}^2 - \frac{1}{4} \text{CL}^2 \right) = m.nf.d^2 \cdot \text{QL}^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 3x^2}{2} \right) \); but QL^2 = b^2 - x^2; therefore \( \frac{1}{4} \text{QL}^2 - \frac{1}{4} \text{CL}^2 = \frac{1}{4}b^2 - \frac{1}{4}x^2 - x^2 = \frac{1}{4}b^2 - \frac{3}{4}x^2 = \frac{b^2 - 3x^2}{2} \); therefore the momentum of the ring is \( m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 3x^2}{2} \right) = m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 4b^2x^2 + 3x^4}{2} \right) = m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 4b^2x^2 + 3x^4}{2} \right) \).

If we now suppose another parallel extremely near to A a, as represented by the dotted line, the distance L l between them being \( \dot{x} \), we shall have the fluxion of the momentum of the spheroid \( m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 4b^2x^2 + 3x^4}{2} \right) \), of which the fluent is \( m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^2 - 4b^2x^2 + 3x^4}{2} \right) \). This expresses the momentum of the zone EA a Q, contained between the equator and the parallel of latitude A a. Now let \( \dot{x} \) become \( \dot{b} \), and we shall obtain the momentum of the hemispheroid \( m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^5 - 4b^5x^3 + 3x^5}{5} \right) \), and that of the spheroid \( m.nf.d^2 \cdot \text{Pi} \cdot \left( \frac{b^5 - 4b^5x^3 + 3x^5}{5} \right) = \frac{4}{15}m.nf.d^2 \cdot \text{Pi} \cdot b^5 \).

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 \( a^3 \) to \( b^3 \), and the contents of the inscribed sphere is \( \frac{2}{3} \text{Pi} \cdot b^3 \). Therefore \( a^3 : a^3 - b^3 = \frac{2}{3} \text{Pi} \cdot b^3 : \frac{2}{3} \text{Pi} \cdot b^3 \), which is the quantity of protuberant matter. We may, without sensible error, suppose \( \frac{a^2 - b^2}{a} = 2d \); then the protuberant matter will be \( \frac{4}{3} \pi b^2 d \). If all this were placed at O, the momentum would be \( \frac{4}{3} \pi db^2 f.OH.HC \), because OH.HC = mnfd; now \( \frac{4}{3} \) is 5 times \( \frac{1}{3} \).

Also, because the sum of all the rectangles OH.HC round the equator is half of as many squares of OC, it follows that the momentum of the protuberant matter placed in a ring round the equator of the sphere or spheroid is one half of what it would be if collected in the point O or E; whence it follows that the momentum of the protuberant matter in its natural place is two-fifths of what it would be if it were disposed in an equatorial ring. It was in this manner that Sir Isaac Newton was enabled to compare the effect of the sun's action on the protuberant matter of the earth, with his effect on a rigid ring of moons. The preceding investigation of the momentum is nearly the same with his, and appears to us greatly preferable in point of perspicuity to the fluxionary solutions given by later authors. These indeed have the appearance of greater accuracy, because they do not suppose all the protuberant matter to be condensed on the surface of the inscribed sphere: nor were we under the necessity of doing this, only it would have led to very complicated expressions had we supposed the matter in each line AQ collected in its centre of oscillation or gyration. We made a compensation for the error introduced by this, which may amount to \( \frac{1}{15} \) of the whole, and should not be neglected, by taking \( d \) as equal to \( \frac{a^2 - b^2}{2a} \) instead of \( \frac{a^2 - b^2}{a + b} \).

The consequence is, that our formula is the same with that of the later authors.

Thus far Sir Isaac Newton proceeded with mathematical rigour; but in the application he made two assumptions, or, as he calls them, hypotheses, which have been found to be unwarranted. The first was, that when the ring of protuberant matter is connected with the inscribed sphere, and subjected to the action of the disturbing force, the same quantity of motion is produced in the whole mass as in the ring alone. The second was, that the motion of the nodes of a rigid ring of moons is the same with the mean motion of the nodes of a solitary moon. But we are now able to demonstrate, that it is not the quantity of motion, but of momentum, which remains the same, and that the nodes of a rigid ring move twice as fast as those of a single particle. We proceed therefore to

Prob. 2. To determine the deviation of the axis, and the retrograde motion of the nodes which result from this libratory momentum of the earth's protuberant matter.

But here we must refer our readers to some fundamental propositions of rotatory motions which are demonstrated in the article Rotation.

If a rigid body is turning round an axis A, passing through its centre of gravity with the angular velocity \( a \), and receives an impulse which alone would cause it to turn round an axis B, also passing through its centre of gravity, with the angular velocity \( b \), the body will now turn round a third axis C, passing through its centre of gravity, and lying in the plane of the axes A and B, and the sine of the inclination of this third axis to the axis A will be to the sine of inclination to the axis B as the velocity \( b \) to the velocity \( a \).

When a rigid body is made to turn round any axis 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 \( \frac{4}{3} \pi a^2 b^2 v \).

The two last are to be found in every elementary book of mechanics.

Let AN an (fig. 4.) be the plane of the earth's equator, cutting the ecliptic CNK n in the line of the nodes or equinoctial points N n. Let OAS be the section of the earth by a meridian passing through the sun, so that the line OCS is in the ecliptic, and CA is an arch of an hour-circle or meridian, measuring the sun's declination. The sun not being in the plane of the equator, there is, by prop. 1, a force tending to produce a libration round an axis ZO \( \approx \) at right angles to the diameter A a of that meridian in which the sun is situated, and the momentum of all the disturbing forces is \( \frac{4}{3} \pi mnfd \Pi b^2 \). The product of any force by the moment of its action expresses the momentary increment of velocity; therefore the momentary velocity, or the velocity of libration generated in the time \( i \) is \( \frac{4}{3} \pi mnfd \Pi b^2 i \). This is the absolute velocity of a point at the distance \( i \) from the axis, or it is the space which would be uniformly described in the moment \( i \), with the velocity which the point has acquired at the end of that moment. It is double the space actually described by the libration during that moment; because this has been an uniformly accelerated motion, in consequence of the continued and uniform action of the momentum during this time. This must be carefully attended to, and the neglect of it has occasioned very faulty solutions of this problem.

Let \( v \) be the velocity produced in the point A, the most remote from the axis of libration. The momentum excited or produced in the spheroid is \( \frac{4}{3} \pi a^2 b^2 v \) (as above), and this must be equal to the momentum of the moving force, or to \( \frac{4}{3} \pi mnfd \Pi b^2 i \); therefore we obtain \( v = \frac{4}{3} \pi mnfd \Pi b^2 i \), that is, \( v = mnfd \Pi b^2 \) or very nearly \( mnfd \), because \( \frac{b^2}{a^2} = 1 \) very nearly. Also, because the product of the velocity and time gives the space uniformly described in that time, the space described by A in its libration round Z \( \approx \) is \( mnfd \), and the angular velocity is \( \frac{mnfd}{a} \).

Let \( r \) be the momentary angle of diurnal rotation. The arch A \( r \), described by the point A of the equator in this moment \( i \) will therefore be \( ar \), that is, \( a \times r \), and the velocity of the point A is \( \frac{ar}{i} \), and the angular velocity of rotation is \( \frac{r}{i} \).

Here then is a body (fig. 5.) turning round an axis Fig. 5. Precession. OP, perpendicular to the plane of the equator \( z \circ x \), and therefore situated in the plane \( ZP \circ \); and it turns round this axis with the angular velocity \( \frac{r}{i} \). It has received an impulse, by which alone it would librate round the axis \( Z \circ \), with the angular velocity \( \frac{m n f d i}{a} \). It will therefore turn round neither axis (No. 31.), but round a third axis \( OP' \), passing through O, and lying in the plane \( ZP \circ \), in which the other two are situated, and the fine \( P'N \) of its inclination to the axis of libration \( Z \circ \) will be to the fine \( Pp \) of its inclination to the axis \( OP \) of rotation as \( \frac{r}{i} : \frac{m n f d i}{a} \).

Now A, in fig. 4, is the summit of the equator both of libration and rotation; \( m n f d i \) 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 \( Ac \) (perpendicular to the plane of the equator of rotation, and lying in the equator of libration,) as \( ar \) to \( mnfdi \), and completing the parallelogram \( Armc \), \( Am \) will be the compound motion of \( A \) (No. 31.), and \( ar : mnfdi = 1 : \frac{mnfdi}{ar} \), which will be the tangent of the angle \( mAr \), or of the change of position of the equator.

But the axes of rotation are perpendicular to their equator; and therefore the angle of deviation \( w \) is equal to this angle \( rAm \). This appears from fig. 5.; for \( P'P : P'P = OP : OP' = OP : tan. POP \); and it is evident that \( ar : mnfdi = \frac{mnfdi}{ar} \), as is required 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 \( Nn \). 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 \( An \) of the former equator is above the new one \( An' \); 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 and down like the arm of a balance. On this account Precession, this motion is very properly termed libration; but this very slow libration, compounded with the incomparably swifter motion of diurnal rotation, produces a third motion extremely different from both. At first the north pole of the earth inclines forward toward the sun; after a long course of years it will incline to the left hand, 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 of a mariner's compass, having its centre of gravity coinciding with the point of the cap, so that it may whirl round in any position. As this is extremely difficult to execute, the cap may be pierced a little deeper, which will cause the ring to maintain a horizontal position with a very small force. When the ring is whirling very steadily, and pretty briskly, in the direction 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 likewise; and the equator, instead of being raised or depressed round its east and west points, is twisted likewise 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 Newton; 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 earth did in this manner deviate from its former position, still, however, turning round its axis of figure. In this he has been followed by Walmelly, 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 Friesius 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 Friesius 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.

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 Timocharis, 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 latitudes 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, instead of being at P, which we may suppose to be a particular mountain, is now at p. In this 12 hours the mountain P, by its rotation round PR, has acquired the position π. At the end of the next 12 hours, the axis of rotation has got the position π, and the axis of figure 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 \( \frac{1}{365} \) 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.

We have hitherto considered this problem in the most general manner; let us now apply the knowledge we have gotten of the deviation of the axis or of the momentary action of the disturbing force to the explanation of the phenomena: that is, let us see what precession and 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.

The gravitation of the earth to the sun is in the proportion of the sun's quantity of matter M directly, and to the square of the distance A inversely, and may therefore be expressed by the symbol \( \frac{M}{A^2} \). The disturbing force at the distance r from the plane of illumination is to the gravitation of the earth's centre to the sun as 3 to A, (A being measured on the same scale which measures the distance from the plane of illumination).

Therefore \( \frac{3M}{A^3} \) will be the disturbing force \( f \) of our formula.

Let \( p \) be the centrifugal force of a particle at the distance \( r \) from the axis of rotation; and let \( t \) and \( T \) be the times of rotation and of annual revolution, viz. sidereal day and year. Then \( p = \frac{M}{A^2} = \frac{r^3}{T^2} \). Hence we derive \( \frac{3M}{A^3} = \frac{3p}{T^2} \). But since \( r \) was the angular velocity of rotation, and consequently \( r \times r \) the space described, and \( \frac{r}{t} \) the velocity; and since the centrifugal force is as the square of the velocity divided by the radius, (this being the measure of the generated velocity, which is the proper measure of any accelerating force), we have \( p = \frac{r^3}{T^2} \times \frac{r^3}{T^2} \), and \( f = \frac{3r^3}{T^2} \times \frac{r^3}{T^2} \).

Now the formula \( f = mn \frac{d}{a} \) expresses the sine of the angle. This being extremely small, the sine may be considered as equal to the arc which measures the angle. Now, substitute for it the value now found, viz. \( \frac{3r^3}{T^2} \times \frac{r^3}{T^2} \), and we obtain an angle of deviation \( w = \frac{3r^3}{T^2} \times \frac{r^3}{T^2} \), and this is the simplest form in which it can appear. But it is convenient, for other reasons, to express it a little differently: \( d \) is nearly equal to \( \frac{a^2 - b^2}{2a^2} \), therefore \( w = \frac{3r^3}{T^2} \times \frac{r^3}{T^2} \times \frac{a^2 - b^2}{a^2} \), and this is the form in which we shall now employ it.

The small angle \( \frac{3r^3}{T^2} \times \frac{a^2 - b^2}{a^2} \) is the angle in which the new equator cuts the former one. It is different at different times, as appears from the variable part \( mn \), the product of the sine and cosine of the sun's declination. It will be a maximum when the declination is in the solstice, for \( mn \) increases all the way to 45°, and the declination never exceeds 23°. It increases, therefore, from the equinox to the solstice, and then diminishes. Let ESL (fig. 7.) be the ecliptic, EAC the equator, BAD the new position which it acquires by the momentary action of the sun, cutting the former in the angle BAE = \( \frac{3}{2} \frac{T^2}{m n} \). Let S be the sun's place in the ecliptic, and AS the sun's declination, the meridian AS being perpendicular to the equator. Let \( \frac{a^2 - b^2}{a^3} \) be k. The angle BAE is then \( \frac{3}{2} \frac{T^2}{k m n} \). In the spherical triangle BAE we have fin. B : fin. AE = fin. A : fin. BE, or = AB : BE, because very small angles and arches are as their fines. Therefore BE, which is the momentary precession of the equinoctial point E, is equal to \( \frac{\text{fin. AE}}{\text{fin. B}} = \frac{3}{2} \frac{T^2}{k m n} \).

The equator EAC, by taking the position BAD, recedes from the ecliptic in the colour of the solstices CL, and CD is the change of obliquity or the nutation. For let CL be the solstitial colour of BAD, and c the solstitial colour of EAC. Then we have fin. B : fin. E = fin. LD : fin. LC; and therefore the difference of the arches LD and LC will be the measure of the difference of the angles B and E. But when BE is indefinitely small, CD may be taken for the difference of LD and LC, they being ultimately in the ratio of equality. Therefore CD measures the change of the obliquity of the ecliptic, or the nutation of the axis with respect to the ecliptic.

The real deviation of the axis is the same with the change in the position of the equator, Pp being the measure of the angle EAB. But this not being always made in a plane perpendicular to the ecliptic, the change of obliquity generally differs from the change in the position of the axis. Thus when the sun is in the solstice, the momentary change of the position of the equator is the greatest possible; but being made at right angles to the plane in which the obliquity of the ecliptic is computed, it makes no change whatever in the obliquity, but the greatest possible change in the precession.

In order to find CD the change of obliquity, observe that in the triangle CAD, R : fin. AC, or R : cof. AE = fin. A : fin. CD, = A : CD (because A and CD are exceedingly small). Therefore the change of obliquity (which is the thing commonly meant by nutation) CD = A × cof. AE = \( \frac{3}{2} \frac{T^2}{k m n} \), cof. AE' = \( \frac{3}{2} \frac{T^2}{k} \times \text{fin. declin.} \times \text{cof. declin.} \times \text{cof. R asecn.} \).

But it is more convenient for the purposes of astronomical computation to make use of the sun's longitude SE. Therefore make

- The sun's longitude ES = \( \frac{3}{2} \frac{T^2}{k m n} \) - Sine of sun's long. = \( \frac{3}{2} \frac{T^2}{k m n} \) - Cofine = \( \frac{3}{2} \frac{T^2}{k m n} \) - Sine obliqu. eclipt. = \( \frac{3}{2} \frac{T^2}{k m n} \) - Cofine obliq. = \( \frac{3}{2} \frac{T^2}{k m n} \)

In the spherical triangle EAS, right-angled at A (because AS is the sun's declination perpendicular to the equator), we have R : fin. ES = fin. E : fin. AS, and fin. AS = px. Also R : cof. AS = cof. AE : cof. ES, and cof. ES or y = cof. AS × cof. AE. Therefore Precession \( p x y = \text{fin. AS} \times \text{cof. AS} \times \text{cof. AE} = mn \times \text{cof. AE} \).

Therefore the momentary mutation CD = \( \frac{3}{2} \frac{T^2}{k} \times \frac{3}{2} \frac{T^2}{k} \times p x y \).

We must recollect that this angle is a certain fraction of the momentary diurnal rotation. It is more convenient to consider it as a fraction of the sun's annual motion, so we may directly compare his motion on the ecliptic with the precession and nutation corresponding to his situation in the heavens. This change is easily made, by augmenting the fraction in the ratio of the sun's angular motion to the motion of rotation, or multiplying the fraction by \( \frac{T}{t} \); therefore the momentary nutation will be \( \frac{3}{2} \frac{T^2}{k} \times p x y \). In this value \( \frac{3}{2} \frac{T^2}{k} \) is a constant quantity, and the momentary nutation is proportional to \( x y \), or to the product of the fine and cofine of the sun's longitude, or to the fine of twice the sun's longitude; for \( x y \) is equal to half the fine of twice \( x \).

If therefore we multiply this fraction by the sun's momentary angular motion, which we may suppose, with abundant accuracy, proportional to \( x \), we obtain the fluxion of the nutation, the fluent of which will express the whole nutation while the sun describes the arch \( x \) of the ecliptic, beginning at the vernal equinox. Therefore in place of \( y \) put \( \sqrt{1-x^2} \), and in place of \( x \) put \( \frac{x}{\sqrt{1-x^2}} \), and we have the fluxion of the nutation for the moment when the sun's longitude is \( x \), and the fluent will be the whole nutation. The fluxion resulting from this process is \( \frac{3}{2} \frac{T^2}{k} \times x \dot{x} \), of which the fluent is \( \frac{3}{4} \frac{T^2}{k} \times x^2 \). This is the whole change produced on the obliquity of the ecliptic while the sun moves along the arch \( x \) ecliptic, reckoned from the vernal equinox. When this arch is \( 90^\circ \), \( x^2 \) is \( r \), and therefore \( \frac{3}{4} \frac{T^2}{k} \) is the nutation produced while the sun moves from the equinox to the solstice.

The momentary change of the axis and plane of the equator (which is the measure of the changing force) is \( \frac{3}{2} \frac{T^2}{k} \times \dot{x} \).

The momentary change of the obliquity of the ecliptic is \( \frac{3}{4} \frac{T^2}{k} \times \dot{x} \).

The whole change of obliquity is \( \frac{3}{4} \frac{T^2}{k} \times x^2 \).

Hence we see that the force and the real momentary change of position are greatest at the solstices, and diminish to nothing in the equinoxes.

The momentary change of obliquity is greatest at the octants, being proportional to \( x \dot{x} \) or to \( x y \).

The whole accumulated change of obliquity is greatest at the solstices, the obliquity itself being then smallest.

We must in like manner find the accumulated quantity quantity of precession in a given time.

We have \( EB : CD = \tan \) fin. EA : fin. CA (or cof. EA) = tan. EA : 1, and EB : ER = 1 : fin. B. Therefore \( EB : CD = \tan \) fin. long. \( = \frac{q}{\sqrt{1-x^2}} \).

Therefore \( EB : CD = \frac{q}{\sqrt{1-x^2}} p \), and \( CD = EB : \) fin. obliqu. eclipt. \( \circ \). If we now substitute for CD its value found in No. 40 viz. \( \frac{3tkp}{2T} \), we obtain \( EB = \frac{3tk}{2T} \times \frac{kq x^2}{\sqrt{1-x^2}} \), the fluxion of the precession of the equinoxes occasioned by the action of the sun. The fluent of the variable part \( \frac{x^2}{\sqrt{1-x^2}} = xy \), of which the fluent is evidently a segment of a circle whose arch is \( z \) and fine \( x \), that is, \( \frac{z-x\sqrt{1-x^2}}{2} \), and the whole precession, while the sun describes the arch \( z \), is \( \frac{3tk}{2T} \left( 2 - x\sqrt{1-x^2} \right) \). This is the precession of the equinoxes while the sun moves from the vernal equinox along the arch \( z \) of the ecliptic.

In this expression, which consists of two parts, \( \frac{3tk}{4T} z \) and \( \frac{3tk}{4T} \left( -x\sqrt{1-x^2} \right) \), the first is incomparably greater than the second, which never exceeds \( 1'' \), and is always compensated in the succeeding quadrant. The precession occasioned by the sun will be \( \frac{3tk}{4T} z \), and from this expression we see that the precession increases uniformly, or at least increases at the same rate with the sun's longitude \( x \), because the quantity \( \frac{3tk}{4T} \) is constant.

In order to make use of these formulae, which are now reduced to very great simplicity, it is necessary to determine the values of the two constant quantities \( \frac{3tk}{4T}, \frac{3tk}{4T} \), which we shall call N and P, as factors of the nutation and precession. Now \( t \) is one sidereal day, and \( T \) is 366.5. \( k \) is \( \frac{a^2-b^2}{a^2} \), which according to Sir Isaac Newton is \( \frac{23^2-230^2}{23^2} = \frac{1}{115} \); \( p \) and \( q \) are the sine and cosine of \( 23^\circ 28' \), viz. 0.39822 and 0.91729.

These data give \( N = \frac{1}{141030} \) and \( P = \frac{1}{61224} \), of which the logarithms are 4.85069 and 5.21308, viz. the arithmetical complements of 5.14931 and 4.78692.

Let us, for an example of the use of this investigation, compute the precession of the equinoxes when the sun has moved from the vernal equinox to the summer solstice, so that \( z = 90^\circ \), or 324000''. Precession. the angle E contained between the equator and the lunar orbit, the precession will be \( \frac{m \pi t}{T} \cdot \text{Cof. E} \), and it must be reckoned on the lunar orbit.

Now let \( \varphi B \) (fig. 8.) be the immovable plane of the ecliptic, \( \varphi ED = F \) the equator in its first situation, before it has been deranged by the action of the moon, AGRDBH the equator in its new position, after the momentary action of the moon. Let EGNFH be the moon's orbit, of which N is the ascending node, and the angle \( N = 5^\circ 8' 46'' \).

Let \( N \varphi \) the long. of the node be

- Sine \( N \varphi \) - Cosine \( N \varphi \) - Sine \( \varphi = 23^\circ \) - Cosine \( \varphi \) - Sine \( N = 5.8.46 \) - Cosine \( N \) - Circumference to radius \( r = 6.28 \) - Force of the moon - Solar precession (supposed \( = 14.7'' \) by observation) - Revolution of \( D = 27^\circ \) - Revolution of \( O = 366^\circ \) - Revolution of \( N = 18 \) years \( 7 \) months

In order to reduce the lunar precession to the ecliptic, we must recollect that the equator will have the same inclination at the end of every half revolution of the sun or of the moon, that is, when they pass through the equator, because the sum of all the momentary changes of its position begins again each revolution. Therefore if we neglect the motion of the node during one month, which is only \( 1^\circ \) degrees, and can produce but an inensible change, it is plain that the moon produces, in one half revolution, that is, while she moves from H to G, the greatest difference that she can in the position of the equator. The point D, therefore, half-way from G to H, is that in which the moveable equator cuts the primitive equator, and DE and DF are each \( 90^\circ \). But S being the foreshortened point, \( \varphi S \) is also \( 90^\circ \). Therefore \( DS = \varphi E \). Therefore, in the triangle DGE, we have \( \text{fin. } ED : \text{fin. } G = \text{fin. } EG : \text{fin. } D = \text{EG : D. Therefore } D = \text{EG} \times \text{fin. } G, = \text{EG} \times \text{fin. } E \) nearly. Again, in the triangle \( \varphi DA \) we have \( \text{fin. } A : \text{fin. } \varphi D \) (or \( \text{cof. } \varphi E \)) \( = \text{fin. } D : \text{fin. } \varphi A = D : \varphi A \). Therefore \( \varphi A = \frac{D \cdot \text{cof. } \varphi E}{\text{fin. } A}, = \frac{\text{EG} \cdot \text{fin. } E \cdot \text{cof. } \varphi E}{\text{fin. } 23^\circ} = \frac{m \pi t}{T} \cdot \text{fin. } E \cdot \text{cof. } \varphi E \).

This is the lunar precession produced in the course of one month, estimated on the ecliptic, not constant like the solar precession, but varying with the inclination or the angle E or F, which varies both by a change in the angle N, and also by a change in the position of N on the ecliptic.

We must find in like manner the nutation SR produced in the same time, reckoned on the colure of the solstices RL. We have \( R : \text{fin. } DS = D : RS \), and \( RS = D \cdot \text{fin. } DS, = D \cdot \text{fin. } \varphi E \). But \( D = \text{EG} \cdot \text{fin. } E \). Therefore \( RS = \text{EG} \cdot \text{fin. } E \cdot \text{fin. } \varphi E, = \frac{m \pi t}{T} \cdot \text{cof. } \varphi E \times \text{fin. } E \times \text{fin. } \varphi E \). In this expression we must substitute

the angle N, which may be considered as constant during the month, and the longitude \( \varphi N \), which is also nearly constant, by observing that \( \text{fin. } E : \text{fin. } \varphi N = \text{fin. } N : \text{fin. } \varphi E \). Therefore \( RS = \frac{m \pi t}{T} \times \text{fin. } N \cdot \text{fin. } \varphi N \cdot \text{cof. } \varphi E \).

But we must exterminate the angle E, because it changes by the change of the position of N. Now, in the triangle \( \varphi EN \) we have \( \text{cof. } E = \text{cof. } \varphi N \cdot \text{fin. } N \cdot \text{fin. } \varphi - \text{cof. } N \cdot \text{cof. } \varphi, = y \cdot c \cdot a - d \cdot b \). And because the angle E is necessarily obtuse, the perpendicular will fall without the triangle, the cofine of E will be negative, and we shall have \( \text{cof. } E = b \cdot d - a \cdot c \cdot y \). Therefore the nutation for one month will be \( \frac{m \pi t}{T} \times \frac{c \cdot x \cdot (b \cdot d - a \cdot c \cdot y)}{b} \), the node being supposed all the while in N.

These two expressions of the monthly precession and nutation may be considered as momentary parts of the fluxions of the whole variable precession and nutation, while the node continually changes its place, and in the space of 18 years makes a complete tour of the heavens.

We must, therefore, take the motion of the node as the fluent of comparison, or we must compare the fluxions and nutations of the node's motion with the fluxions of the precession and nutation; therefore, let the longitude of the node be \( x \), and its monthly change \( \dot{x} \); we shall then have \( t = \frac{n \cdot \dot{x}}{e} \), and \( t = \frac{n \cdot \dot{x}}{e} \cdot \frac{c}{a \sqrt{1-x^2}} \). Let T be \( = 1 \), in order that \( n \) may be 18.6, and substitute for \( t \) its value in the fluxion of the nutation, by putting \( \sqrt{1-x^2} \) in place of \( y \). By this substitution we obtain \( m \pi n \cdot \frac{c}{e \cdot b} \left( \frac{d \cdot b \cdot x \cdot \dot{x}}{\sqrt{1-x^2}} - a \cdot c \cdot x \cdot \dot{x} \right) \). The fluent of this is \( m \pi n \cdot \frac{c}{e \cdot b} \left( -d \cdot b \cdot \sqrt{1-x^2} - a \cdot c \cdot x^2 \right) \). (Vide Simpson's Fluxions, § 77.) But when \( x \) is \( = 0 \), the nutation must be \( = 0 \), because it is from the position in the equinoctial points that all our deviations are reckoned, and it is from this point that the period of the lunar action recommences. But if we make \( x = 0 \) in this expression, the term \( -a \cdot c \cdot x^2 \) vanishes, and the term \( -d \cdot b \cdot \sqrt{1-x^2} \) becomes \( -d \cdot b \); therefore our fluent has a constant part \( +d \cdot b \); and the complete fluent is \( m \pi n \cdot \frac{c}{e \cdot b} \left( d \cdot b - d \cdot b \cdot \sqrt{1-x^2} - a \cdot c \cdot x^2 \right) \). Now this is equal to \( m \pi n \cdot \frac{c}{e \cdot b} \left( d \cdot b \times \text{versed fine}, \frac{1}{x} \cdot a \cdot c \times \text{versed fine} \times 2 \right) \): For the versed fine of \( x \) is equal to \( (1 - \text{cof. } x) \); and the square of the fine of an arch is \( \frac{1}{4} \) the versed fine of twice that arch.

This, then, is the whole nutation while the moon's ascending node moves from the vernal equinox to the longitude \( \varphi N = x \). It is the expression of a certain number of seconds, because \( \pi \), one of its factors, is the solar precession in seconds; and all the other factors are numbers, or fractions of the radius \( r \); even \( e \) is expressed in terms of the radius \( r \).

The fluxion of the precession, or the monthly precession, Precession, is to that of the nutation as the cotangent of \( \varphi E \) is to the fine of \( \varphi \). This also appears by considering fig. 7. \( P p \) measures the angle \( A \), or change of position of the equator; but the precession itself, reckoned on the ecliptic, is measured by \( P o \), and the nutation by \( p o \); and the fluxion of the precession is equal to the fluxion of nutation \( \times \cot. \varphi E \), but \( \cot. \varphi E = \frac{a + b c y}{c x} \); therefore \( \cot. \varphi E = \frac{ad + bc}{c x} \sqrt{\frac{1 - x^2}{x}} \): This, multiplied into the fluxion of the nutation, gives \( \frac{m \pi n}{a b e} \left( \frac{a d^2}{\sqrt{1 - x^2}} + (b^2 - a^2) d c - ab c \sqrt{1 - x^2} \right) \) for the monthly precession. The fluent of this \( \frac{m \pi n}{a b e} \left( ad^2 b x + (b^2 - a^2) d c x - \frac{1}{2} ab c \sqrt{1 - x^2} \right) \), or it is equal to \( \frac{m \pi n}{a b e} \left( (d^2 - \frac{1}{2} c^2) ab x + (b^2 - a^2) d c x - \frac{1}{2} ab c \sqrt{1 - x^2} \right) \).

Let us now express this in numbers: When the node has made a half revolution, we have \( z = 180^\circ \), whose versed sine is 2, and the versed sine of \( 2z \), or \( 360^\circ \), is 0; therefore, after half a revolution of the node, the nutation (\( N^\circ 52 \)) becomes \( \frac{m \pi n c}{e b} \times 2bd \). If, in this expression, we suppose \( m = 24 \), and \( \pi = 14^\circ \), we shall find the nutation to be \( 19^\circ 3'' \).

Now the observed nutation is about \( 18'' \). This requires \( m \) to be \( 27^\circ \), and \( \pi = 16^\circ \). But it is evident that no astronomer can pretend to warrant the accuracy of his observations of the nutation within \( 1'' \).

To find the lunar precession during half a revolution of the node, observe that then \( z \) becomes \( \frac{e}{2} \), and the fine of \( z \) and of \( 2z \) vanish, \( d^2 \) becomes \( 1 - c^2 \), and the precession becomes \( \frac{m \pi n}{2} \left( d^2 - \frac{1}{2} c^2 \right) = \frac{m \pi n}{2} \left( 1 - \frac{1}{2} c^2 \right) \), and the precession in 18 years is \( m \pi n \times \frac{1}{2} c^2 \).

We see, by comparing the nutation and precession for nine years, that they are as \( \frac{4 cd}{c} \) to \( 1 - \frac{1}{2} c^2 \), nearly as 1 to \( 17^\circ \). This gives \( 313'' \) of precession, corresponding to \( 18'' \), the observed nutation, which is about \( 35'' \) of precession annually produced by the moon.

And thus we see, that the inequality produced by the moon in the precession of the equinoxes, and, more particularly, the nutation occasioned by the variable obliquity of her orbit, enables us to judge of her share in the whole phenomenon; and therefore informs us of her disturbing force, and therefore of her quantity of matter. This phenomenon, and those of the tides, are the only facts which enable us to judge of this matter: and this is one of the circumstances which has caused this problem to occupy so much attention. Dr Bradley, by a nice comparison of his observations with the mathematical theory, as it is called, furnished him by Mr Machin, found that the equation of precession computed by that theory was too great, and that the theory would agree better with the observations, if an ellipse were substituted for Mr Machin's little circle. He thought that the shorter axis of this ellipse, lying in the colure of the solstices, should not exceed \( 16'' \). Nothing can more clearly show the astonishing accuracy of Bradley's observations than this remark: for it results from the theory, that the pole must really describe an ellipse, having its shorter axis in the solstitial colure, and the ratio of the axis must be that of 18 to 16.8; for the mean precession during a half revolution of the node is \( \frac{m \pi n}{2} \left( d^2 - \frac{1}{2} c^2 \right) \); and therefore, for the longitude \( x \), it will be \( \frac{m \pi n}{e} \left( d^2 - \frac{1}{2} c^2 \right) \); when this is taken from the true precession for that longitude (\( N^\circ 54 \)), it leaves the equation of precession \( \frac{m \pi n}{a b e} \left( (b^2 - a^2) d c \sin x - \frac{1}{2} ab c \sin 2x \right) \); therefore, when the node is in the solstice, and the equation greatest, we have it \( \frac{m \pi n c}{a b e} \left( b^2 - a^2 \right) \). We here neglect the second term as insignificant.

This greatest equation of precession is to \( \frac{2 m \pi n c d}{c} \), equation of the nutation of \( 18'' \), as \( b^2 - a^2 \) to \( 2ab \); that is, radius to the tangent of twice the obliquity of the ecliptic. This gives the greatest equation of precession \( 16'' \), not differing half a second from Bradley's observations.

Thus have we attempted to give some account of this curious and important phenomenon. It is curious, because it affects the whole celestial motions in a very intricate manner, and received no explanation from the more obvious application of mechanical principles, which so happily accounted for all the other appearances. It is one of the most illustrious proofs of Sir Isaac Newton's sagacity and penetration, which catches at a very remote analogy between this phenomenon and the libration of the moon's orbit. It is highly important to the progress of practical and useful astronomy, because it has enabled us to compute tables of such accuracy, that they can be used with confidence for determining the longitude of a ship at sea. This alone fixes its importance; but it is still more important to the philosopher, affording the most incontestable proof of the universal and mutual gravitation of all matter to all matter. It left nothing in the solar system unexplained from the theory of gravity but the acceleration of the moon's mean motion; and this has at last been added to the list of our acquisitions by M. de la Place.

Quae toties animos veterum tortere Sophorum, Quaeque scholas frustra rauco certamine vexant, Obvia conspicimus, nube pellente Mathei, Jam dubios nulla caligine praegravat error Quae superum penetrae domos, atque ardua coeli Scandere fulmis genii concepsit acumen. Nec fas est propius mortali attingere divos.

Halley.

PRECLÆ, (precious, "early"), the name of the 21st order in Linnæus's fragments of a natural method; consisting of primrose, an early flowering plant, and a few few genera which agree with it in habit and structure, though not always in the character or circumstance expressed in the title. See Botany, Natural Orders.

**PRECEITANT**, in Chemistry, is applied to any liquor, which, when poured on a solution, separates what is dissolved, and makes it precipitate, or fall to the bottom of the vessel.

**PRECEITATE**, in Chemistry, a substance which, having been dissolved in a proper menstruum, is again separated from its solvent, and thrown down to the bottom of the vessel by pouring some other liquor upon it.

**PRECEITATION**, the process by which a precipitate is formed.