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S has moved through about half the quadrant (more correctly till S1 is 46° 14' 10" from the equinox), and then diminishes till S arrives at the tropic or equinox towards which it is moving, and that the greatest difference in time thus occasioned is about 9m 531.
We have thus two sources of inequality; one arising from the unequal motion of the sun in his orbit, the other from the inclination of that orbit to the equator. To ascertain how the time at which S comes to the meridian of the place, or the true solar time, differs from that at which S, comes to the same meridian, or the mean solar time, we must combine the two. To see how this combination is to be effected, let us begin from the winter solstice, and examine what will be the relative positions of S and S for the year: the difference in time between the instants of their being on the meridian is called the equation of time, being the difference between the actual time of the sun's being on the meridian and the beginning of the mean solar day t. The sun is in perigee about the 30th of December. We have already seen, that in passing from apogee to perigee, S, is always more advanced, or comes later on the meridian, than S: and consequently, at the winter solstice, which is a little before the sun comes to perigee, S, is more advanced than S. But at the solstices S, and S. come on the meridian together, and consequently, S, is then also more advanced than S, or the mean solar time is later than the apparent, or the sun is on the meridian before noon by the clock, or the clock is after the sun. From the solstice to the equinox S2 continually is less advanced than S., but at perigee, S. and S coincide, therefore at perigee S2 is less advanced than S, or the mean solar time is earlier than the
The equation of time is only registered from noon to noon, when S and S2 are on the meridian; but the term applies with equal correctness to the difference of time between the instants of their
arrival at any given distance from the meridian. It is plain that this is correctly an equation, being the difference of the real and mean times of the oc
currence of a particular event.
The mean solar day is considered, for astronomical purposes, to begin when S2 is on the meridian, or at noon. The period of its commencement may of course be arbitrarily fixed, and for astronomical purposes this is the most convenient; for civil purposes that is the most convenient which includes all the active period of day-light within one day, and the civil day, therefore, begins at midnight, when the sun is on the meridian below the horizon, or midway between his setting and rising. Each astronomical day, therefore, contains the last twelve hours of one, and the first twelve of the next, civil day.
apparent, or the clock before the sun : and as a few days before it was later, and all the changes are gradual and continuous, there must have been some intermediate instant when S and S, were on the meridian together, or the mean and apparent time coincided, or the equation of time was nothing. This would be on December 24th, 1829; and from that time till the perigee, the mean would be earlier than the apparent time, or the clock before the sun. From the perigee onwards, S, is continually less advanced than S; and from the solstice to the equinox S, is continually less advanced than S1. On both accounts therefore, during the whole period from the perigee to the vernal equinox S, is less advanced than S, or the mean solar time is earlier than the apparent, or the clock is before the sun. After the vernal equinox until the apogee, S continues more advanced than Si; but from the equinox to the solstice S is more advanced also than S, when therefore these differences are equal, the equation of time becomes nothing. Now, the greatest difference in time between the approach of S and S, to the meridian never amounts to more than 8m 24,; the greatest difference between the approach of S, and S. to the meridian is of about 9m 53s, and takes place a little more than half way between the equinox and the solstice, or about May 8th. By May 8th therefore, the difference between S, and S. has become greater than that between S and S., having been less at the equinox, and between these times the differences must have been equal, or S and S. must have been on the meridian together, or the equation of time must have been nothing. In point of fact, it is so on April 15th, 1830. After this time, S, and S are both more advanced than S1, but S, more so than S; or S is more advanced than S, or the mean time later than the apparent, or the clock after the sun. At the summer solstice however, S and S, are again together: but the apogee does not take place till after the summer solstice (on June 30th), therefore S continues more advanced than S1, or, at the summer solstice, S, which on April 16th was less advanced than S., has become more so, and there will therefore have been an instant when they came on the meridian together, and the equation of time was nothing. This is on June 15th, 1830. From that time to the solstice, it is clear that S is more ad
vanced than S,, or the apparent later than the mean time, or the clock before the sun. From the solstice to the autumnal equinox, S, is continually more advanced than S2, but until the apogee S is more advanced than S1, and of course, than Sg: for this period therefore, the apparent time is later than the mean time, or the clock before the sun. After the apogee however, S is continually less advanced than S1, and as at the autumnal equinox S, and S. are together, S, which at the apogee was more advanced than S2, has then become less so, and there has been some intervening instant at which S and S2 have come on the meridian together, or the equation of time has been nothing. This is on September 1st, 1830; and from that time to the equinox S is less advanced than S., and the apparent is earlier than the mean time, or the clock is after the sun. After the equinox, until the winter solstice, the sun is still moving from apogee towards perigee, and S is consequently less advanced than S1, and in the interval between equinox and solstice, S, is also less advanced than S: on both accounts therefore S is less advanced, during this whole period, than S2, or the apparent time is earlier than the mean time, or the clock after the sun.
We have now gone through the year, and we may collect our results thus: that in the course of the year there are four days, and only four, namely, December 24th, April 15th, June 15th, and Sept. 1st, when the apparent and mean time are the same, or the equation of time is nothing: and that in the interval between the first and second of these, and again in that between the third and fourth, the apparent is always later than the mean time, or the clock before the sun: and that between the second and third, and again between the fourth and first, the apparent is always earlier than the mean time, or the clock after the sun. These results correspond with those in the common tables of the equation of time.
It is also evident, from the manner in which these results have been deduced, that they depend entirely on the relative positions of the apogee, and of the equinoxes. If these are fixed points, or hold always the same relative position, the results we have obtained will serve alike for every year: if they vary, the equation of time will vary also; and this consideration leads us to inquire whether there be any motion of the equinoxes,
and whether the apogee and perigee be or be not fixed points. As far also as the magnitude of the equation is concerned, it is evident that any variation in the inclination of the ecliptic to the equator would affect it; for the angle
S, s (in fig. 6), and the declination of the sun at any point of the ecliptic, would both be affected by this change; and both these quantities are involved in the solution of that part of the question which arises from the motion of the sun in a plane inclined to the equator.
In point of fact, it is found that the inclination of the ecliptic and equator does undergo some slight variation. This is not sufficient to produce any material alteration in the results, or to call for more extended notice here; but it furnishes one reason why the results obtained for the equation of time cannot, as far as their numerical values are concerned, apply accurately, except to the particular periods for which they are computed. The other considerations are more important in themselves, and will deserve separate consideration. They are also of practical importance, as connected with the division of time into longer periods than we have yet used, except in a loose and popular way of speaking. Our next section therefore will treat of the precession of the equinoxes, and the progression of the apogee; the following one of the length of the year, and the consequent corrections and adjustment of the calendar.
SECTION 8.-Precession of the Equinoxes produced by the retrograde motion of the Equator on the Ecliptic-Effects on the longitude, latitude, declination, and right ascension of the Heavenly Bodies-Progressive motion of the Sun's apogee and perigee.
In speaking of the mode of ascertaining the declination of a heavenly body, we have only referred to observations made on the meridian. And they are the best adapted to that purpose generally; it may, however, also be computed from the altitude and azimuth* of the object when observed out of the meridian. But when we take the case
The azimuth of a body is the arc of the horizon intercepted between the meridian and a verof a body with respect to a particular place on the tical circle passing through the body. The situation earth, is determined by its altitude and azimuth, just as its situation, with respect to the heavens in ascension or by its latitude and longitude. general, is detera ired by its declination and right
of a body whose declination, like that of the sun, continually varies, it is clear that we cannot be sure of ascertaining the time at which his declination is of any given value by observations on the meridian; for he is upon the meridian and above the horizon, only for one instant (or in some cases two), in twentyfour hours; and there is no greater likelihood that his declination will then be that concerning which we inquire, than at any other instant in that time. Nor can we even be sure of ascertaining the required time by the means mentioned at the beginning of this section, for he may attain the required declination while he is below the horizon, and then no direct observation can be made. We are not however without the means of ascertaining the required period. The sun's change of declination, although very different in amount at different periods of his course, may with little error be considered as uniform during the small space of twenty-four hours; and we consequently have the means, by observations made on the meridian on two successive days, of computing his declination at particular times in the interval between them. Thus, if at noon on September 20th the sun's declination were 11' North, and at noon on the following day 13' South, we might safely estimate his change of declination at 1' hourly (the whole difference in the twentyfour hours being of 24'); and we should therefore say, that at six o'clock in the evening of September 20th, his declination was 5' North; and that it was nothing, or that the equinox took place, at eleven o'clock on the same evening. For purposes of greater accuracy, there are easy means of making the same computations, allowing for the variation of the rate of the sun's change of declination.
gistered, or obtained by computation from the corresponding right ascensions and declinations. The same process may be gone through at a subsequent time. If this be done, we shall find the right ascensions and declinations of all stars altered, but in various manners and degrees, the latitude of all remaining very nearly the same, and the longitude of all increased by very nearly the same quantity. We shall also find that the different stars keep the same position with respect to each other; they continue at the same distance from, and make the same angles with, each other. The alteration therefore in their right ascensions, &c. of which we have spoken, does not proceed from any motion among themselves; it must proceed from some alteration in some of the arbitrarily assumed points or lines from which these elements are measured, and by which they are estimated. And these are the equinoxes, and the equator and ecliptic.
Having thus the means of ascertaining the time at which the sun is at any particular declination, and consequently, among others, the time of the equinox, we find, especially by comparing to gether the observations made at distant times, that when the sun now comes to the meridian at the equinoxes, the same stars are not on the meridian which formerly were so when he came thither at that part of his annual course. The position of the equinox being ascertained at a particular time, a catalogue of stars can be formed, and their right ascensions and declinations registered. Their longitudes and latitudes may also be re
Let us first take the simpler class of phenomena. We have already stated that the longitude of every heavenly body is increased by the same quantity, and that their latitude is not affected. This would evidently be the case if the position of the ecliptic itself continued unaltered, but the point from which arcs are measured along it were removed backward; for the latitude, being the perpendicular distance between the star and the fixed circle, remains the very same arc, the longitude, being the distance from the same point (the intersection of the secondary passing through the star with the ecliptic) to the arbitrary standard whence the measurement is taken, is increased exactly as much as that standard is removed. The longitude and latitude therefore are affected as they would be if the ecliptic remained unmoved itself, but the first point of Aries receded upon it.
The first point of Aries however is the intersection of the ecliptic with the equator: and if this is moved, while the ecliptic remains stationary, it can only be so by moving the whole circle of the equator. The inclination of one of these circles to the other is always found to be very nearly the same: the distance of their poles is always equal to the inclination, and it therefore is always the same. If then the equinox moves backwards gradually, the equator is continually changing its position; its pole he refore is continually moving, but still
always at the same distance from that of the ecliptic; and consequently the pole of the equator would describe a circle or part of a circle about that of the ecliptic. The position of any given star continuing fixed, the line of the secondary to the equator which passes through it would vary, for it passes through the star and the pole of the equator, and this latter point has changed its position; the position of the equator itself would vary also, and so also would that of the first point of Aries. The declination therefore would vary; for the pole of the equator, moving while the star continued fixed, would generally approach towards or recede from it; and the North polar distance being thus altered, and the North polar distance and the declination together making 90°, the declination would vary also. This variation also would be different for different stars; for the same motion which brings the pole nearer to some carries it further from others; and this at a different rate, as its motion is directly to wards or from the star, or oblique with respect to it. In the same manner the right ascension of different stars would be differently affected. All would have a certain effect produced on them by the alteration of the point from which the measurement is taken: but this would not be the only cause of alteration. The right ascension is the distance from the first point of Aries to the intersection with the equator of a secondary to that circle passing through the star. As these secondaries all pass through the pole of the equator, the old and new secondaries would intersect each other at the star, and make different angles with each other in the case of different stars, as those stars are situated nearer to or farther from the poles, and in one direction or another with respect to them. Intersecting each other at the star, they would diverge after they passed it, and, consequently, even if the angles made were the same in two different instances, the space by which the secondaries would have separated before meeting the equator, would be different as the distance of the stars from that circle, or the declination, differed. The variation therefore of right ascension would be different on all these accounts in different stars. These different results and variations may be with advantage illustrated by a figure, as far as relates to the right ascensions and declinations. The positions with respect to the longitude and latitude are too
simple to require such illustration, and the figure would be inconveniently complicated by the introduction of the circles relating to them.
Let tT (in fig. 7) represent one position of the equator, op't' T' another at some considerable distance of time, and let P & P' be the corresponding positions Fig. 7.
of the pole. Let S and s be any two stars, and through S draw PST, P'ST' secondaries to the first and second positions of the equator respectively; and in like manner Pst, P'st', through s. is evident that ST, T, represent the declination and right ascension of the star S, when the equator is in the position T, and that ST', 'T', represent them when the equator is in the position op't' T'. In the same manner, st, cp t, are the declination and right ascension of the star s in the first position of the equator; st', 't', in the second. If SV be taken equal to SP, and s v equal to s P', it is evident from inspection of the figure that SV is less than SP', and sv than s P. In the former case therefore the new North polar distance is greater than the old one, or the declination has diminished: in the latter the new North polar distance is less than the old one, or the declination has increased.
We next proceed to examine the variation in right ascension. For this purpose let us draw through, the former position of the equinox, P'op X, a secondary to the new position of the equator. On the new equator therefore the right ascension of is p'X, and the right ascensions of all the heavenly bodies are increased by this quantity. This however is not the only variation. The remaining portion of the new right ascension, X T', or X t', is obviously not necessarily equal to T, or opt. In the figure as drawn it is greater in each case; it may in other positions of the star be less. But in every case it
is evident that the difference between them will depend upon the magnitude of the angle TST or tst', or, which is the same thing, on that of the angles PS P', Ps P', and also on the distance of S and s from the circles op tT, op' t' T'. It will therefore vary for every star: and this is all which we are here desirous of illustrating.
We have thus seen generally what would be the nature of the effect produced by the retrocession of the equator upon the ecliptic: that the latitudes of stars would remain the same; that their longitudes would be increased uniformly; that their declinations would be differently affected, some being increased, others diminished, and this in unequal amounts and proportions; and that the right ascensions also, although with many exceptions (in cases where the quantity X T' is less than o T, and their difference exceeds 'X) would generally increase, but at different rates in different cases*. We have seen also that these phenomena are actually observed to take place; and we therefore lay it down as an established fact, that while the ecliptic continues immoveable, the equator has a retrograde motion upon it, or a motion from left to right in those already referred to. The amount of this, subject to some small inequalities, is 50".1 in a year; that is to say, the first point of Aries recedes annually 50".1 upon the ecliptic: the retrocession, therefore, is 1° 23′ 30′′ in a century, or a degree in about 71
It is not desirable here to introduce the calculations on which the results depend; but the results themselves may be given with advantage. The precession indeclination is found to be positive, that is to say, the declination is increased by the effect of precession, wherever the right ascension of the star is less than 90°, or greater than 270°; the precession is negative, or the declination is diminished when the right ascension is between 90° and 270: it is nothing, or the declination is not affected when the right ascension is 90° or 270°. The angle formed by a secondary to the equator and a secondary to the ecliptic, each passing through the star, is called the angle of position of the star. The precession in right ascension is positive, or the right as cension is increased wherever the angle of position is less than 90°; it is nothing, when that angle is 90°; it is negative, or the right ascension is dimi
nished, when that angle is greater than 90°.
These are the effects of a small variation in the position of the equinoctial points, or of the precession for a short period. The effects of precession for a long time, when they become considerable, must be deduced from computation of the accu mulated effects of these minute variations; for the right ascensions being continually changed by the effect of precession, its effect on declination, which and the pole of the equator changing its place, the angle of position, which is determined by it, will vary also, and thus the variation of right ascension will also itself be changeable.
depends on them, will continually chauge also;
years; and the first point of Aries will have receded through the whole circle, and consequently will return to its present position in about 25868 years. This retrograde motion is called, on account of its effects on the time of the occurrence of the equinox, which it accelerates, the precession of the equinoxes, i. e., their going forward.
So slow an alteration may seem of little importance, except in a very long series of years. In a science however, where none except the most accurate results are of practical value and importance, no cause of error is to be neglected; and especially where, as in the present case, the error is of a nature continually accumulating. Thus the difference occasioned in the equation of time by this alteration of the position of the equinox, would at present be but slight, and generally only affect the actual numerical value of that correction for a considerable period; but we have already seen that the nature of the correction is mainly dependent on the relative positions of the equinoxes or solstices and the sun's apogee or perigee. The perigee is now nearly 10° more advanced, or has 10° greater longitude, than the winter solstice: in the year 1250, it coincided with the winter solstice, and before that time it preceded it; the combined operation of the retrocession of the equinox on the ecliptic, and a progressive motion of the perigee itself, having since then brought them into their present relative positions. those times therefore the equation of time would not only differ in amount from its present values, but the considerations used in deducing the periods at which the apparent is before or after the mean time, would themselves differ.
The motion which we have just mentioned to exist in the perigee, or apogee, (for as the two are always 180° distant from each other, they must move alike) is also deduced from observation. If the very instant of the sun's being in perigee or apogee could be readily de termined, this motion would easily be would be determined, and the 'alteration ascertained; for his place at the time of that place, when he was next in the like situation, would be the motion required. The variations however of his apparent diameter, or of his angular motion, by which alone we can immediately estimate those of his distance, are too slow to admit of any very accurate estimation of very small