multiplying the fraction by \( \frac{T}{T} \); therefore the momentary nutation will be \( \frac{3t}{2T} k \rho \times y \). In this value \( \frac{3t}{2T} k \rho \) is a constant quantity, and the momentary nutation is proportional to \( xy \), or to the product of the sine and cosine of the sun's longitude, or to the sine of twice the sun's longitude; for \( xy \) is equal to half the sine of twice \( z \).

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

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

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

Hence we see that the force and the real momentary changes 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 \( x \) or to \( y \).

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 of the precession after a given time, that is, the arch BE for a finite time.

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

Therefore EB : CD \( = \frac{7x}{\sqrt{1-x^2}} \rho \), and CD = EB : fin. obliq. eclipt. tan. long. \( \circ \).

If we now substitute for CD its value found in \( n^\circ 40 \), viz. \( \frac{3t}{2T} k \rho \times x \), we obtain EB \( = \frac{3t}{2T} \times \frac{k \rho \times x}{\sqrt{1-x^2}} \), the fluxion of the precession of the equinox. Precession equinoxes occasioned by the action of the sun. The fluent of the variable part \( \frac{x^2}{\sqrt{1-x^2}} = x \dot{y} \), 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 \( \frac{3t}{2T} \left( \frac{kq}{4T} (2 - x \sqrt{1-x^2}) \right) \). 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, \( \frac{3tkq}{4T} \), and \( \frac{3tkq}{4T} (-x \sqrt{1-x^2}) \), 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 \( \frac{3tkq}{4T} \), and from this expression we see that the precession increases uniformly, or at least increases at the same rate with the sun's longitude \( z \), because the quantity \( \frac{3tkq}{4T} \) 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 \( \frac{3tkp}{4T}, \frac{3tkq}{4T} \), which we shall call \( N \) and \( P \), as factors of the nutation and precession. Now \( t \) is one sidereal day, and \( T \) is \( 366^\circ \). \( k \) is \( \frac{a^2-b^2}{a^2} \), which according to Sir Isaac Newton is \( \frac{231^\circ - 230^\circ}{231^\circ} = \frac{1}{115} \); \( p \) 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 \( 324000'' \).

Log \( 324000 = z = 5.51035 \) Log \( P = 5.21308 \) Log \( 5'' = 0.72363 \)

The precession therefore in a quarter of a year is \( 5,292 \) seconds; and, since it increases uniformly, it is \( 21''168 \) annually.

We must now recollect the assumptions on which this computation proceeds. The earth is supposed to be homogeneous, and the ratio of its equatorial diameter to its polar axis is supposed to be that of \( 231 \) to \( 230 \). If the earth be more or less protuberant at the equator, the precession will be greater or less in the ratio of this protuberance. The measures which have been taken of the degrees of the meridian are very inconsistent among themselves; and although a comparison of them all indicates a smaller protuberance, nearly \( \frac{1}{115} \) instead of \( \frac{1}{117} \), their differences are too great to leave much confidence in this method. But if this figure be thought more probable, the precession will be reduced to about \( 17'' \) annually. But even though the figure of the earth were accurately determined, we have no authority to say that it is homogeneous. If it be denser towards the centre, the momentum of the protuberant matter will not be so great as if it were equally dense with the inferior parts, and the precession will be diminished on this account. Did we know the proportion of the matter in the moon to that in the sun, we could easily determine the proportion of the whole observed annual precession of \( 50'' \) which is produced by the sun's action. But we have no unexceptionable data for determining this; and we are rather obliged to infer it from the effect which she produces in disturbing the regularity of the precession, as will be considered immediately. So far, therefore, as we have yet proceeded in this investigation, the result is very uncertain. We have only ascertained unquestionably the law which is observed in the solar precession. It is probable, however, that this precession is not very different from \( 20'' \) annually; for the phenomena of the tides show the disturbing force of the sun to be very nearly \( \frac{7}{8} \) of the disturbing force of the moon. Now \( 20'' \) is \( \frac{7}{8} \) of \( 50'' \).

But let us now proceed to consider the effect of the moon's action on the protuberant matter of the earth; the moon's and as we are ignorant of her quantity of matter, and action on consequently of her influence in similar circumstances with the sun, we shall suppose that the disturbing force of the moon is to that of the sun as \( m \) to \( 1 \). Then earth. (ceteris paribus) the precession will be to the solar precession \( \pi \) in the ratio of the force and of the time of its action jointly. Let \( t \) and \( T \) therefore represent a periodical month and year, and the lunar precession will be \( \frac{m \cdot t}{T} \). This precession must be reckoned on the plane of the lunar orbit, in the same manner as the solar precession is reckoned on the ecliptic. We must also observe, that \( \frac{m \cdot t}{T} \) represents the lunar precession only on the supposition that the earth's equator is inclined to the lunar orbit in an angle of \( 23^\circ \) degrees. This is indeed the mean inclination; but it is sometimes increased to above \( 28^\circ \), and sometimes reduced to \( 18^\circ \). Now in the value of the solar precession the cosine of the obliquity was employed. Therefore whatever is the angle \( E \) contained between the equator and the lunar orbit, the precession will be \( \frac{m \cdot t}{T} \cdot \text{Cof. } E \) and it must be reckoned on the lunar orbit.

Now let \( \gamma B \) (fig. 8.) be the immovable plane of the ecliptic, \( \gamma ED \) 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 \( LGNFH \) be the moon's orbit, of which \( N \) is the ascending node, and the angle \( N = 5^\circ 8' 46'' \).

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

| Sine \( N \gamma \) | \( z \) | --- | --- | | Coline \( N \gamma \) | \( y \) | Sine \( \gamma = 23^\circ \) | \( a \) | Cosine \( \gamma \) | \( b \) | Sine \( N = 5.8.46 \) | \( c \) | Cosine \( N \) | \( d \) | Circumference to radius \( r = 6.28 \) | \( e \) | Force | \( f \) Force of the moon Solar precession (supposed = 14½" by observation) Revolution of C = 27½ Revolution of C = 366½ 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 ecliptic, the sun or 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 1½ degrees, and can produce but an insensible 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°. But S being the solstitial point, VS is also 90°. Therefore DS = VE. Therefore, in the triangle DGE, we have fin. ED : fin. G = fin. EG : fin. D, = EG : D. Therefore D = EG × fin. G, = EG × fin. E nearly. Again, in the triangle γDA we have fin. A : fin. γD (or cof. γE) = fin. D : fin. γA, = D : γA. Therefore

\[ \frac{D}{\sin \gamma A} = \frac{EG}{\sin \gamma E} \cdot \frac{\sin \gamma A}{\sin \gamma E} = \frac{m}{T} \cdot \frac{\sin E \cdot \text{Cof. } E \cdot \text{Cof. } \gamma E}{\sin \gamma \cdot \text{Cof. } \gamma}. \]

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 parallaxes RL. We have R : fin. DS = D : RS, and RS = D : fin. DS = D : fin. VE. But D = EG × fin. E. Therefore RS = EG × fin. E × fin. VE = \( \frac{m}{T} \cdot \text{Cof. } \gamma \times \text{fin. } E \times \text{fin. } VE \).

In this expression we must substitute the angle N, which may be considered as constant during the month, and the longitude γN, which is also nearly constant, by observing that fin. E : fin. γN = fin. N : fin. VE. Therefore RS = \( \frac{m}{T} \times \frac{\sin N \cdot \sin \gamma N \cdot \text{Cof. } E}{\text{Cof. } \gamma} \).

But we must exterminate the angle E, because it changes by the change of the position of N. Now, in the triangle γEN we have cof. E = cof. γN × fin. N × fin. γ = cof. N × cof. γ, = yca = acy. And because the angle E is necessarily obtuse, the perpendicular will fall without the triangle, the cosine of E will be negative, and we shall have cof. E = b - acy. Therefore the nutation for one month will be \( \frac{m}{T} \times \frac{cy(bd - acy)}{b} \), 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 moon's action, corresponding to a certain position of the node and inclination of the equator, or as 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 of the node's motion with the fluxions of the precession and nutation; therefore, let the longitude of the node be z, and its monthly change \( \frac{dz}{dt} \); we shall then have

\[ t = n = z, \quad \text{and} \quad \frac{dz}{dt} = \frac{n}{c \sqrt{1 - z^2}}. \]

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 \( \sqrt{1 - z^2} \) in place of \( y \). By this substitution we obtain \( m \times n \times \frac{c}{e \times b} \)

\[ \left( \frac{db \times x}{\sqrt{1 - x^2}} - ac \times x \right). \]

The fluent of this is \( m \times n \times \frac{c}{e \times b} \)

\[ \left( - db \sqrt{1 - x^2} - ac \times x \right). \] (Vide Simpson's Fluxions, § 77). But when \( x = 0 \), the nutation must be zero, because it is from the position in the equinoctial points that all our deviations are reckoned, and it is from this point that the periods of the lunar action recommences. But if we make \( x = 0 \) in this expression, the term \( - ac \times x \) vanishes, and the term \( - db \sqrt{1 - x^2} \) becomes \( - db \); therefore our fluent has a constant part \( + db \);

and the complete fluent is \( m \times n \times \frac{c}{e \times b} \left( db - db \sqrt{1 - x^2} - ac \times x \right) \).

Now this is equal to \( m \times n \times \frac{c}{e \times b} \left( db \times \text{versed fine } z \right) \): For the versed sine of \( z \) is equal to \( (1 - \text{cosine } z) \); and the square of the line of an arch is \( \frac{1}{2} \) the versed sine of twice that arch.

This, then, is the whole nutation while the moon's ascending node moves from the vernal equinox to the longitude γN = z. It is the expression of a certain number of seconds, because \( z \), one of its factors, is the solar precession in seconds; and all the other factors are numbers, or fractions of the radius 1; even \( c \) is expressed in terms of the radius 1.

The fluxion of the precession, or the monthly precession, is to that of the nutation as the cotangent of γE is to the fine of γE. This also appears by considering figure 7. PP measures the angle A, or change of position of the equator; but the precession itself, reckoned on the ecliptic, is measured by PE, and the nutation by PO; and the fluxion of the precession is equal to the fluxion of nutation \( \times \frac{\cot \gamma E}{\text{fine } \gamma E} \), but \( \cot \gamma E = \frac{a + b \times c}{a \times c} \); therefore \( \frac{\cot \gamma E}{\text{fine } \gamma E} = \frac{a + b \times c \sqrt{1 - x^2}}{a \times c} \): This, multiplied into the fluxion of the nutation, gives \( m \times n \times \frac{a \times d}{a \times e} \left( \frac{bd}{\sqrt{1 - x^2}} + \frac{(b^2 - a^2)}{a \times c} \times \frac{dc}{a \times c} \times \frac{ab}{a \times c} \times \sqrt{1 - x^2} \right) \times \text{for the monthly precession.} \)

The fluent of this \( m \times n \times \frac{a \times d}{a \times e} \left( \frac{bd}{\sqrt{1 - x^2}} + \frac{(b^2 - a^2)}{a \times c} \times \frac{dc}{a \times c} \times \frac{ab}{a \times c} \times \sqrt{1 - x^2} \right) \), or it is equal to \( m \times n \times \frac{a \times d}{a \times e} \left( \frac{(d^2 - c^2)}{a \times c} \times abz + \frac{(b^2 - a^2)}{a \times c} \times dc \times \frac{ab}{a \times c} \times \text{fine } 2z \right) \). Let us now express this in numbers: When the node has made a half revolution, we have \( z = 180^\circ \), whose versed sine is \( z \), and the versed sine of \( 2z \), or \( 360^\circ \), is \( = 0 \); therefore, after half a revolution of the node, the nutation (no 52.) becomes \( \frac{m \times n}{c} \times \frac{e}{b} \times d \). If, in this expression, we suppose \( m = z \), and \( r = 14 \frac{1}{2} \), we shall find the nutation to be \( 19 \frac{1}{2} \).

Now the observed nutation is about \( 18'' \). This requires \( m \) to be \( 2 \frac{1}{2} \), and \( r = 16 \frac{1}{2} \). 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 sine of \( z \) and of \( 2z \) vanish, \( d^2 \) becomes \( 1 - c^2 \), and the precession becomes \( \frac{m \times n}{2} \left( d^2 - \frac{1}{2} c^2 \right) = \frac{m \times n}{2} \left( 1 - \frac{1}{2} c^2 \right) \), and the precession in \( 18 \) years is \( m \times n \times \frac{1}{2} \times \frac{1}{2} c^2 \).

We see, by comparing the nutation and precession for nine years, that they are as \( \frac{4}{c} \) to \( 1 - \frac{1}{2} c^2 \) nearly as \( 1 \) to \( 17 \frac{1}{2} \). 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 axes must be that of \( 18 \) to \( 16 \frac{1}{2} \); for the mean precession during a half revolution of the node is \( \frac{m \times n}{2} \left( d^2 - \frac{c^2}{2} \right) \); and therefore, for the longitude \( z \), it will be \( \frac{z \times m \times n}{c} \left( d^2 - \frac{c^2}{2} \right) \); when this is taken from the true precession for that longitude (no 54.), it leaves the equation of precession \( \frac{m \times n}{a \times b \times e} \left( b^2 - a^2 \right) \times c \times \text{fine } z - \frac{1}{2} a \times b \times c \times \text{fine } 2z \); therefore, when the node is in the solstice, and the equation greatest, we have it \( = \frac{m \times n \times c \times d}{a \times b \times e} \left( b^2 - a^2 \right) \). We here neglect the second term as insignificant.

This greatest equation of precession is to \( \frac{z \times m \times n \times c \times d}{c} \), the nutation of \( 18'' \), as \( b^2 - a^2 \) to \( 2ab \); that is, as 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 Mr de la Place.

Quae toties animos veterum torseret Sophorum, Quaeque scholas fruatur raucos certamine vexant, Obvia confincimus, nube pelletae Mathesi, Jam dubios nulla caligine praegratam error Quae superum penetrare domos, atque ardua coeli Scandere sublimis genii concepit acumen. Nec fas est propius mortali attingere divos.

Halley.

PRECIÆ, (precious, "early,") the name of the first order in Linnæus's fragments of a natural method; consisting of primrose, an early flowering plant, and a few genera which agree with it in habit and structure, though not always in the character or circumstance expressed in the title. See Botany, p. 461, col. 2.