PRECESSION OF THE EQUINOXES

PRECESSION OF THE EQUINOXES. The most obvious of all the celestial motions is the diurnal revolution of the starry heavens. The whole appears to 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 $\backslash alpha$ 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 and night, and the vicissitudes of the seasons. By a 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 ECLIPTIC; 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 occupying these stations; for it was natural to compute the course of the year from that moment. Accordingly this has been the leading problem in the astronomy of 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 between the years 161 and 127 before Christ, Hipparchus,

Precession. chas, 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 Aristillus and Timochares, 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 asserts 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 $20^\{\backslash circ\}\; 21\text{'}$ westward of Spica Virginis. Supposing the motion to have been uniform during this period of ages, it follows that the annual precession is about $50\backslash frac\{1\}\{2\}\text{'}\text{'}$; 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 $50\backslash frac\{1\}\{2\}\text{'}\text{'}$ to the west of it, and the sun will come to the equinox $20^\{\backslash circ\}\; 23\text{'}\text{'}$ 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 this 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 $23\backslash frac\{1\}\{2\}$ degrees in 25,745 years; and in the time of Timochares, 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 saros 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 forewarner of the overflowing of the Nile; and they combined him with the star Fomalhast * in their mystical calendar. This knowledge is also involved in the 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, 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 every thing 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 any thing more. Had he known the Indian precession of $54\text{'}\text{'}$ 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 Timochares, 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 could be assigned. But the establishment of the Copernican system reduced it to a very simple affair; the 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 $23^\{\backslash circ\}\; 56\text{'}\; 4\text{'}\text{'}$, which axis is inclined to this plane in an angle of nearly $23^\{\backslash circ\}\; 28\text{'}$; 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

Precession. the distance of $23^\{\backslash circ\}\; 28\text{'}$ 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 $50\backslash frac\{1\}\{2\}$ 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.

¹¹ Bradley's attempts to discover the parallax of the earth's orbit. 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.

¹² Difficulties in the attempt obviated by accident. 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.

¹³ His further investigation of the subject. 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 $18\text{'}\text{'}$, 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 Macleod, 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 $23^\{\backslash circ\}\; 28\text{'}$, 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, whose diameter AC is $18\text{'}\text{'}$. Mathematically, 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 node. Draw EPF and GPL perpendicular to it; let GL 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 $90^\{\backslash circ\}$ 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 $18\text{'}\text{'}$, and the shorter, BD, $16\text{'}\text{'}$. Mr d'Alembert determined, by the physical theory of gravitation, the axes to be $18\text{'}\text{'}$ and $13\text{'}\text{'}$.

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 arc 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 $g\text{'}\text{'}$ when the moon's ascending node is in the vernal equinox, and consequently the pole in A. It is, on the contrary, diminished $g\text{'}\text{'}$ 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 arc ADCO is equal to the node's longitude from the vernal equinox, and

cession, 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 $=\; 9\text{'}\text{'}\; \backslash times\; \backslash cos.\; \backslash text\{\; long.\; node\}$, 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 changes also the longitudes and right ascensions of the stars and planets, by changing the equinoctial points, and thus occasioning an equation in the precession of the equinoctial points. 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 S 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 $M\text{'}EM\text{'}\text{'}$, $mEm\text{'}$, perpendicular to PE, OE.

If the pole were in its mean place P, the equinoctial points would be in the ecliptic meridian $M\text{'}EM\text{'}\text{'}$, or that meridian would pass through the intersections of the equator and ecliptic, and the angle $M\text{'}ES$ would measure the longitude of the star S. But when the pole is in O, the ecliptic meridian $mEm\text{'}$ 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 $mES$, 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 $\backslash frac\{OM\}\{OE\}$.

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 $9\text{'}\text{'}\; \backslash times\; \backslash text\{long.\; node\}$, and $\backslash frac\{OM\}\{OE\}$ is equal to $\backslash frac\{9\text{'}\text{'}\; \backslash times\; \backslash text\{long.\; node\}\}\{\backslash sin.\; \backslash text\{\; obliq.\; eclip.\}\}$. 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 six signs, and additive to it while the node is in the last six signs.

This equation of longitude is the same for all the stars, for the 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.

their. When the pole is at P, the right ascension of 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 $x\; =\; \backslash frac\{9\text{'}\text{'}\; \backslash times\; \backslash sin.\; \backslash text\{\; long.\; node\}\}\{\backslash text\{tang.\; obl.\; eclip.\}\}$. This is subtractive from the mean right ascension for the first six signs of the node's longitude, and additive for the last six signs. This equation is common to all the stars.

The variation of the other part SOG of the angle, 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 Cotesian theorem, $\backslash tan.\; SP\; :\; \backslash sin.\; SPG\; =\; PO\; :\; y$, and $y$, or the second part of the nutation in right ascension, $=\; 9\text{'}\text{'}\; \backslash times\; \backslash sin.\; \backslash text\{\; diff.\; R.\; A.\; of\; star\; and\; node\}$.

The nutation also affects the declination of the stars: For SP, the mean codeclination, is changed into SO. Suppose a circle described round S, with the distance SO cutting SP in $f$; then it is evident that the equation of declin. is $Pf\; =\; PO\; \backslash times\; \backslash cosine\; \backslash \; OPf\; =\; 9\text{'}\text{'}\; \backslash times\; \backslash sin.\; \backslash text\{\; ascen.\; of\; star\}\; \backslash times\; \backslash text\{long.\; of\; node\}$.

Such are the calculations in constant use in our astronomical researches, founded on Machin's Theory. When still greater accuracy is required, the elliptical theory must be substituted, by taking (as is expressed by the dotted lines) O in that point of the ellipse described on the transverse axis AC, where it is cut by OM, drawn according to Machin's Theory. All the change made here is the diminution of 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 precession.

Precession. sion, for some hundreds of the principal stars, for every position of the moon's node and of the sun.

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Precession of the equinoctial points, &c. 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.

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Observations of Newton and others on this subject. 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 shrewd guess, founded on assumptions, of which it would be extremely difficult to demonstrate either the truth or falsity, and which required the genius of a Newton to pick out in such a complication of abstruse circumstances. The subject has occupied the attention of the first mathematicians of Europe since his time; and is still considered as the most curious and difficult of 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 NMCDn, which cuts the plane of the ecliptic in the line of the nodes Nn, and has one half raised above it, as represented in the figure, the other half being hid below the ecliptic. Suppose this orbit folded down; it will coincide with the ecliptic in the circle N m c d n. Let EX represent the axis of this orbit, perpendicular to its plane, and therefore inclined to the ecliptic. Since the moon gravi-

tates to the sun in the direction MS, which is all above the ecliptic, it is plain that this gravitation has a tendency to draw the moon towards the ecliptic. Suppose this force to be such that it would draw the moon down from M to r in the time that she would have moved from M to t, in the tangent to her orbit. By the combination of these motions, the moon will desert her orbit, and describe the line Mr, which makes the diagonal of the parallelogram; and if no farther action of the sun be supposed, she will describe another orbit M d n', lying between the orbit MCDn 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 Ex, 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 MNm.

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 syzygee, till they get into quadrature, after which it diminishes till the nodes are again in syzygee. 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 syzygee; 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 syzygee, 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-

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 ring 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.

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 inclination of the equator be called $\backslash pi$, and if $t$ be the number of days (sidereal) in a year, the annual motion of

a detached ring will be $360^\backslash circ\; \backslash times\; \backslash frac\{3\backslash sqrt\{t-\backslash pi^2\}\}\{4t\}$. He then shows that the effect of the disturbing force on this ring is to its effect on the matter of the same ring, distributed in the form of an elliptical stratum (but still detached) as 5 to 2; therefore the motion of the nodes

will be $360^\backslash circ\; \backslash times\; \backslash frac\{3\backslash sqrt\{t-\backslash pi^2\}\}\{10t\}$, or $16\text{'}\; 16\text{'}\text{'}\; 24\text{'}\text{'}\text{'}$ 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 quadrantal arch to two squares of a diameter, jointly: Then he shows, that the quantity of matter in the terrestrial sphere is to that in the protuberant matter of the spheroid, as 52900 to 461 (supposing all homogeneous). From these premises it follows, that the motion of $16\text{'}\; 16\text{'}\text{'}\; 24\text{'}\text{'}\text{'}$, must be diminished in the ratio of 10717 to 100, which reduces it to $9\text{'}\; 07\text{'}\text{'}\text{'}$ annually. And this (he says) is the precession of the equinoxes, occasioned by the action of the sun; and the rest of the $501\text{'}\text{'}$ 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. Justly does his celebrated and candid commentator, Daniel Bernoulli, say (in his Dissertation on the Tides, which shared the prize of the French Academy with M^r L'aurin and Euler), that Newton saw through a veil what others could hardly discover with a microscope in the light of the meridian sun. His determination of the form and dimensions of the earth, which is the

foundation of the whole process, is not offered as any Precession thing better than a probable guess, in re difficillima; and it has since been demonstrated with geometrical rigour by M^r L'aurin.

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 $18\backslash frac\{1\}\{2\}$ 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 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.

We proceed now to the examination of this phenomenon upon the fundamental principles of mechanics. Examination of the phenomenon of precession on mechanical principles. Because the mutual gravitation of the particles of matter in the solar system is in the inverse ratio of the squares of the distance, it follows, that the gravitations 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 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 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 inscribed 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 $a$
OC or PC $b$
EO their difference, $=\; \backslash frac\{a^2\; -\; b^2\}\{a\; +\; b\}$ $d$
L | 2 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 $s\; n$ and MT $t\; m$ are more acted on than the segments NT $t\; n$ and MS $s\; m$; 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 $p\; E$. 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.