changes the place of the pole of the equator, leaves its distance from the pole of the ecliptic unaltered; but nutation changes it. The mean place of the pole, P, remaining at the distance of what we must now call the mean obliquity from the pole of the ecliptic, the true place of the pole of the equator, when the ascending node of the moon's orbit coincides with the vernal equinox, is more distant from the pole of the ecliptic, by the space PA, or 9.648, and the obliquity is therefore increased by that amount; and in the same manner, when the ascending node of the moon's orbit coincides with the autumnal equinox, the pole of the equator is at B, and the obliquity is diminished by the same amount, PB, or 9.648. In other positions there will be corresponding alterations; and the nutation which depends on the position of the sun will of course produce effects similar in their nature, but less in amount.

The remaining correction, aberration, will be fully discussed and explained hereafter. Like refraction, it makes the apparent place of a star different from its true place; and it prevents us, except in the case of stars situated in the ecliptic, from ever seeing a heavenly body exactly in its true position or direction. All its effects are produced in the course of a year, and they are different from every different heavenly body, though they follow one law applicable to all. That law may be very shortly stated. In fig. 15, let S Fig. 15.

B

D

represent the true position of any heavenly body, and let ASB be a portion of a circle of latitude (a secondary to the ecliptic) passing through it. The apparent place of the star will always be a point in an ellipse of which the true place of the star, S, is the centre, and the conjugate axis is in the line AB; the transverse axis of course is in the line CD perpendicular to it. If

ACBD, represent this ellipse, S C, the semi-transverse axis, is in all cases 20".246 of a great circle. The semiconjugate axis SA is to the semitransverse axis SC, in the proportion of the sine of the star's latitude to the sine of 90°, or the radius. The ellipse therefore in which each star is to be found, is easily constructed. The apparent place at any particular time is very nearly found by drawing an arc of a great circle through the star to the point of the ecliptic which is 90° behind the place of the sun at the time the intersection of this line with the ellipse described as before pointed out, is the apparent place of the star at the time of observation. We shall hereafter enter into further details with respect to aberration, and deduce from it some very important consequences: for the present this description of it will be sufficient.

We find then that we have no less than five corrections, as they are called, to apply to the place of a heavenly body, as immediately observed, before we can tell its true place: corrections for precession and nutation, which are rather the alteration of the language in which an observation is originally expressed, to make it correspond accurately with the language of a different period, than corrections of any real inaccuracy; corrections for aberration and refraction, to ascertain from the apparent situation of the star the real direction in which it is with respect to the observer; and a correction for diurnal parallax, which need not be applied in the case of the fixed stars, to deduce from the real direction with respect to the observer the true position as estimated with respect to the centre of the earth.

It will probably occur to the reader that there is a want of strict accuracy of reasoning in the processes which we have explained for ascertaining the amount of these corrections, and even the nature of some of them; because in estimating the one, we have not taken the others into consideration. For instance, taking the method explained in page 59, of determining parallax and refraction, we have shewn how the parallax would affect the value of Et, as compared with ET, on the supposition that s was the apparent place of the body, depressed by parallax below its true place S: and in the same manner we have shewn the effect of refraction, by supposing s to be the ap

parent place of the body, raised by refraction above S. If both refraction and parallax operate, it is obvious that the apparent place of the body will be somewhere between s and s', and the same mode of proceeding will not give us the effect of parallax or refraction, but the combined effect of both. We know however from independent considerations, at least very nearly, the manner in which each of these elements varies with the height of the body above the horizon; and we can form a tolerably near, though by no means a sufficient estimate, of the amount of each at a particular height. We can tell that the amount of parallax of a fixed star must be very small indeed, before we can venture to say that it is absolutely too small to be at all estimated: for we know from other considerations, that the refraction varies very nearly as the tangent of the apparent zenith distance, while the parallax varies accurately as the sine of the same angle, or rather of the same angle increased by the amount of the refraction, but we find that the combined effect of the parallax and refraction in the case of the fixed stars, itself varies very nearly as the tangent of the apparent zenith distance; and we therefore conclude, that the parallax must necessarily be very small in comparison with the refraction. We also find that the difference between the angles really observed, and those estimated on the supposition that this refraction varies as the tangent of the apparent zenith distance, does not itself follow the law previously ascertained for the variation of the parallax, or one resembling it; and we consequently infer that this difference does not principally proceed from parallax, and that our supposition of refraction varying according to the variation of the tangent of the apparent zenith distance is itself inaccurate, although it furnishes an approximation to the truth. Knowing then that the parallax, if any, is very small in comparison with the refraction, we know that we may, with little error, consider the whole effect as produced by refraction only and examining the law of refraction on this supposition, we find, (as we have already stated, in the note to p. 50,) that it is more accurately represented as varying, adopting the notation there used, as the tangent of z-3r. Still, if the fixed stars have any observable parallax, however small, it would be included in this quantity, and

the real refraction would be greater than that thus computed, by the amount of that parallax; for the observed place would be above the true one, by the amount, not of the whole refraction, but of the refraction diminished by the parallax. To see whether this is so, we may resort to observations of the moon, whose parallax is considerable. The amount of refraction being very nearly, if not accurately, ascertained, the apparent zenith distance may be corrected from the error occasioned by refraction, or a very near estimate made of the angle ZTS, (in fig. 12.) although our observations are themselves affected by refraction, and therefore do not immediately give that angle. From these observations the amount of parallax may be computed. If it does not vary accurately as the sine of the corrected apparent zenith distance, the angle Z TS, it is evident that the refraction has not yet been rightly computed, but that some material parallax of the fixed stars is involved in the quantity which we have assumed to vary as the tangent of_z 3r: and we should obtain only an approximate value for the parallax. With this approximate value of parallax we might however correct other observations of the same body, made under different circumstances. In these latter observations therefore we should know, approximately at least, how far the apparent place of the body would be higher than that actually observed, if it were not lowered by parallax, or what would be the apparent place of the body, if only affected by refraction; and from this computed place, and the known true place, as ascertained by the mode already referred to, the amount of refraction might again be more accurately ascertained than before, and a new series of results might be thus obtained and compared, so as to furnish a new and yet more accurate estimate, if necessary, of the mode in which refraction varies; and again, having thus a more correct estimate of refraction, we might repeat our observations on parallax and get a more correct estimate of that element also. And a similar process might be repeated alternately, as long as there was any material, irregularity observed, or any discordance between theory and observation. In the case of refraction and parallax, these repeated corrections of one class of observations by another are little necessary, for the effect of

F

parallax on the fixed stars being really quite imperceptible, our first observations on refraction do not require any correction on this account. We have however illustrated the manner in which the law of such corrections may be ascertained by the supposition of the necessity of pursuing the alternate correction further than is really necessary in this case, because the simplicity of the nature of these causes of error and the circumstance that we know independently the general principles which must regulate their variation, makes them furnish a simpler instance than otherwise could have been given of the manner in which such an investigation is to be pursued, and the steps by which errors, in the first instance inevitable, are gradually to be removed.

Where the existence of the correction itself, as in the case of precession, depends entirely on observation, the ascertainment of its exact amount and law is necessarily more difficult; and the number of alternate reductions and adjustments necessary to ascertain its value and the effect of any other source of inaccuracy with which it may be complicated, will be greater; and after all, until some cause in nature is discovered which will account for the being of such an irregularity, its existence may be considered as not completely established. The mode of ascertaining its value however will be of the same nature; and its existence may be assumed as probable, if we find all the phenomena of the heavens corresponding to that supposition. Thus, still taking the case of precession, we have already seen that upon the suppoIsition of the retrograde motion of the equator on the ecliptic by 50".1 annually, the apparent places of every star in the heavens would be affected, but differently according to the different position of each star; and we find that the apparent places of each star do vary in the manner corresponding to that supposition, or very nearly so. It is therefore reasonable to believe that supposition true, even now, while as yet we have no further grounds for thinking it so. is possible that each star may really have a motion which exactly corresponds to the effect which would be produced on its apparent place upon the supposition of this retrograde motion of the equinoxes, and thus that the equinox itself may be stationary; but it is not at all likely that an almost incalculable number of bodies should each of them

It

have 'motions, all different in amount, and many even opposite in direction, and only corresponding to each other when referred to a point arbitrarily chosen with respect to them, and with which they seem naturally to have no more connection than with any other point in the surface of the heavens. Besides, even if these appearances were to be considered as corresponding to real motions in the stars themselves, and not to an alteration of the point and lines with reference to which they are observed, the importance of the correction would not be diminished; for it would then be ascertained that the stars had actually these motions with respect to that point and those lines, and their relative situation therefore would still be altered in the same manner, and our observations would still have to be correctc on account of that alteration, although the name of precession of the equinoxes would no longer be well chosen to express the cause of the necessary correction.

But in truth, whenever we can discover that all heavenly bodies are af fected according to a certain law, that is to say, in a manner corresponding to their condition with reference to some certain object, we may fairly believe that the effect produced depends on their condition with reference to it. Thus if it were merely by observation that we found that parallax varied in all cases as the sine of the apparent zenith distance, and that all bodies, however different the absolute amount of their parallax, had its variations conformed to that rule, we might fairly conclude that the amount of parallax depended on the amount of the zenith distance; and in fact we know from theory that it is so. The same may be said of all the other corrections which have been mentioned: refraction, precession, nutation, aberration, all affect all bodies, however differently in appearance, according to their distances from certain points, lines, or objects; and we therefore conclude that the amount of these several corrections depends on these distances; and that, when we discover their causes, we shall find them such as would make the amount necessarily, and in its very nature, correspond to these distances. We shall hereafter see that it is really so with respect to precession, nutation, and aberration. We already know it with respect to refraction as well as parallax.

WE have yet a further qualification to give to our original statements, with respect to the fixed and invariable position of the stars. We have already seen that the uniform position of which we spoke must not be considered to be that immediately observed, but only that deduced from it, after correction on all the different accounts we have mentioned. It is however found, after this is done, that many of the stars appear not to continue exactly in the same places, but to have a small motion still unexplained and unaccounted for. In this we do not speak of a small class of heavenly bodies, whose motions are very great and well understood, and of which we shall hereafter treat fully under the name of planets, but of many of that large class of bodies, which, from the very slight change which their position undergoes, and the appearance of unmovedness which they consequently present, are known by the name of fixed stars.

It is not necessary here to detail all the precautions necessary in observing these variations; they are very small in amount, the greatest known not being more than 4" in the course of a year, and most of them not exceeding, or even equalling, 1" in that time. It is by the comparison of observations made at distant periods that such minute motions can be best detected. If they accumulate for years, the aggregate will be so considerable that any inaccuracy in its ascertainment will bear but a small proportion to the total amount; and thus, if the whole amount be distributed equally over the whole space of time, the rate of motion will be ascertained, if not with absolute correctness, yet with a very near approach to it. It is thus found that many of the stars appear to have small motions which cannot be referred to any of the causes of variation which we have hitherto mentioned. If any law could be discovered, according to which these motions varied in each particular case, they would only furnish ground for introducing another general correction, to be applied to all stars. But this is not found to be the case: many, indeed far the largest portion of the stars, have not been found to have any such motion, and those which have it, have it of amounts and in directions not only very different in different cases, but the

differences of which cannot be expressed by any assignable law. Thus of two stars situated very near each other, one will appear to have no motion whatever, the other a motion Eastward; while a third, also situated very nearly in the same part of the heavens, may perhaps have a motion directly Westward, or in some other completely different direction.

We cannot refer these apparent motions to any source affecting the accuracy of our observations, for any such would affect all observations, and would affect them in a manner, different perhaps in each particular instance, but corresponding and consistent in all. We therefore are reduced to the conclusion that they are produced by a real motion in the stars in question; or that some of those stars which are called, in common with those which really have no ascertainable motion, fixed stars, have really small continual motions of their own, or, as they are called, proper motions.

These proper motions of upwards of 400 stars, have been ascertained to exceed one-fifth of a second in a year. Many of the more brilliant stars are among those which have these proper motions. We may instance Arcturus, Sirius, Castor, Pollux, Procyon, Regulus, and the first star, or a, of the constellation of the Eagle. The proper motion of Arcturus is such as to diminish his North polar distance, or increase his declination by nearly 2" every year, and to diminish his right ascension by a little more than 1" in the same period. It is evident that these motions are not to be neglected in the use we make of observations connected with his place *.

For this reason the proper motions of several of the more remarkable stars which have them are inserted in astronomical tables. The proper motions, being in all directions, would, independently of their difference in amount, act differently in different cases, upon the right ascension, declination, &c. of the stars, increasing them in some instances, diminishing them in others, and requiring in each case to be separately calculated and estimated. The results of these computations however are themselves registered, or rather they are generally combined with

*We have already seen, (ch. ii. § 1.) instances of the manner in which variations in declination and

right ascension, arising from precession, affect obso in the same manner when they arise from any other cause.

servations. Of course similar variations will do

the effects of precession, under the title of the annual variation of the particular star in question; so that for all purposes of practical utility the observer may at once refer to these computed results, and thus, by a single addition or subtraction, make the necessary correction for the effects both of precession and of the proper motion of the star observed.

We may now, for the present, quit the consideration of the stars. There are other cirumstances connected with them which will hereafter call for examination; but we have now deduced, from their general appearances, all that is necessary to fit us for the consideration of the more complicated phenomena of the moon and the planets; namely, the apparent general daily rotation of the whole sphere of the heavens, and the various corrections and allowances which must be made in every case for the purpose of deducing the true place from that actually observed. In speaking of the moon and planets, we shall always consider these corrections as made, and take no notice of them unless when they incidentally become material for the purpose of determining other elements, as, for instance, the parallax in discovering the distance of any of these objects. For the future therefore, when we speak of the place of a heavenly body, we speak of its place as it would be observed and estimated by an observer at the centre of the earth at the time of making the observation, and after correcting the observation from the errors occasioned by refraction, aberration, and nutation.

Having premised thus much, we proceed to the consideration of the lunar motions.

CHAPTER III.

SECTION I-Motion of the Moon in an Ellipse round the Earth-Motion of the Nodes and Apsides of her Orbit -Periodic time-Synodic period. IT will be unnecessary to go into any detail of the manner in which we ascertain the nature of the moon's motions; they are found by observation of her places at different times, and of her relative distances at those times, just in the same manner as those of the sun. We thus find that the moon, like the sun, moves alternately from North to South, and from South to North, and continually from West to East amongst the fixed stars; and we find that her track, as well as his, may be repre

sented by a great circle of the heavens, or by the intersection of a plane passing through the centre of the earth with the imaginary sphere of the heavens. Observing also her apparent diameter from time to time, we find that this continually increases during one, and continually diminishes through another portion of her course; and comparing these different observations with each other, in the manner already explained in the case of the sun, we find that she also appears to move in an ellipse about the earth, the earth being in one of the foci. By observation of her parallax we find that her mean distance from the earth is about sixty times the earth's radius; the greatest distance of the moon is about sixty-four times, and the least about fifty-six times the radius of the earth, quantities differing from each other in the proportion of 8 to 7: a much higher variation than that which subsists in the solar ellipse. The greatest apparent diameter of the moon is about 33′ 31", the least 29' 22", quantities which obviously are very nearly in that ratio. We also find that her radius vector, like the sun's, describes equal areas in equal times. Another result also, deduced from observations similar to those already mentioned, is, that the apsides of the moon's orbit, the two extremities of the transverse axis, or her perigee and apogee, are not stationary, but move forwards at the mean rate of 40° 40′ 32".2 in every year, or complete a circuit of the heavens in 3232d 13h 56m 16.8.

The orbit of the moon is not in the same plane with that of the sun, it is inclined to it at an angle of little more than 5°; but this inclination (varies: it never falls short of 5°, it sometimes amounts to as much as 5° 18'. The moon therefore is seen sometimes to the North, sometimes to the South of the ecliptic; and as all great circles of a sphere bisect each other, an equal space of her

course in the heavens will be to the North and to the South of that line. The circles intersect each other in only two points; and for the whole space from one to the other, the moon will either be always to the North, or always to the South of the ecliptic. The points where the moon's orbit intersects the plane of the sun's orbit are called the nodes of her orbit; that at which, having been South of the ecliptic, she passes to the Northward of it, is called the ascending node, the point which we have mentioned in treating of nutation; that at which,