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(906). Time, like distance, may be measured by comparison with standards of any length, and all that is requisite for ascertaining correctly the length of any interval, is to be able to apply the standard to the interval throughout its whole extent, without overlapping on the one hand, or leaving unmeasured vacancies on the other; to determine, without the possible error of a unit, the number of integer standards which the interval admits of being interposed between its beginning and end; and to estimate precisely the fraction, over and above an integer, which remains when all the possible integers are subtracted.

(907). But though all standard units of time are equally possible, theoretically speaking, yet all are not, practically, equally convenient. The solar day is a natural interval which the wants and occupations of man in every state of society force upon him, and compel him to adopt as his fundamental unit of time. Its length as estimated from the departure of the sun from a given meridian, and its next return to the same, is subject, it is true, to an annual fluctuation in excess and defect of its mean value, amounting at its maximum to full half a minute. But except for astronomical purposes, this is too small a change to interfere in the slightest degree with its use, or to attract any attention, and the tacit substitution of its mean for its true (or variable) value may be considered as having been made from the earliest ages, by the ignorance of mankind that any such fluctuation existed.

(908). The time occupied by one complete rotation of the earth on its axis, or the mean * sidereal day, may be shewn, on dynamical principles, to be subject to no variation from any external cause, and although its duration would be shortened by contraction in the dimensions of the globe itself, such as might arise from the gradual escape of its internal heat, and consequent refrigeration and shrinking of the whole mass, yet theory, on the one hand, has rendered it almost certain that this cause cannot have effected any perceptible amount of change during the history of the human race; and, on the other, the comparison of ancient and modern observations affords every corroboration to this conclusion. From such comparisons, Laplace has concluded that the sidereal day has not changed by so much as one hundredth of a second since the time of Hipparchus. The mean sidereal day therefore possesses in perfection the essential quality of a standard unit, that of complete invariability. The same is true of the mean sidereal year, if estimated upon an average sufficiently large to compensate the minute fluctuations arising from the periodical variations of the major axis of the earth's orbit due to planetary perturbation (Art 668.).

(909.) The mean solar day is an immediate derivative of the sidereal day and year, being connected with them by the same relation which determines the synodic from the sidereal revolutions of any two planets or other revolving bodies (Art. 418.). The exact determination of the ratio of the sidereal to the solar day, which is a point of the utmost importance in astronomy, is however, in some degree, complicated by the effect of precession, which renders it necessary to distinguish between the absolute time of the earth's rotation on its axis, (the real natural and invariable standard of comparison,) and the mean interval between two successive returns of a given star to the same meridian, or rather of a given meridian to the same star, which not only differs by a minute quantity from the sidereal day, but is actually not the same for all stars. As this is a point to which a little difficulty of conception is apt to attach, it will be necessary to explain it in some detail. Suppose then x the pole of the ecliptic, and P that of the equinoctial, A B C D the solstitial and equinoctial colures at any given epoch, and Ppqr the small circle described by P about i in one revolution of the equinoxes, i. e. in 25870 years, or 9448300 solar days, all projected on the plane of the ecliptic A B C D. Let S be a star anywhere situated on the ecliptic, or between it and the email circle P q r. Then if the pole P were at

* The true sidereal day is variable by the effect of nutation ; but the variation (an excessively minute fraction of the whole) compensates itself in a revolution of the moon's nodes.


rest, a meridian of the earth setting out from P S C, and revolving in the direction C D, will come again to the star after the exact lapse of one sidereal day, or one rotation of the earth on its axis. But P is not at rest After the lapse of one such day it will have come into the situation (suppose) p, the vernal equinox B having retreated to b, and the colure P C having taken up the new position p c Now a conical movement impressed on the axis of rotation of a globe already rotating is equivalent to a rotation impressed

on the whole globe round the axis of the cone, in addition to that which the globe has and retains round its own independent axis of revolution. Such a new rotation, in transferring P to p, being performed round an axis passing through *, will not alter the situation of that point of the globe which has » in its zenith. Hence it follows that pre passing through » will be the position taken up by the meridian PiC after the lapse of an exact sidereal day. But this does not pass through S, but falls short of it by the hour-angle vp S, which is yet to be described before the meridian comes up to the star. The meridian, then, has lost so much on, or lagged so much behind, the star in the lapse of that interval. The same is true whatever be the arc V p. After the lapse of any number of days, the pole being transferred to p, the spherical angle ir p S will measure the total hour angle which the meridian has lost on the star. Now when S lies any where between C and r, thia angle continually increases (though not uniformly), attaining 180° when p comes to r, and still (as will appear by following out the movement beyond r) increasing thence till it attains 360° when p has completed its circuit. Thus in a whole revolution of the equinoxes, the meridian will have lost one exact revolution upon the star, or in 9448300 sidereal days, will have re-attained the star only 9448299 times: in other words, the length of the day measured by the mean of the successive arrivals of any star outside of the circle P p q r on one and the same meridian is to the absolute time of rotation of the earth on its axis as 9448300 : 9448299, or as 1-00000011 to 1.

(910.) It is otherwise of a star situated within this circle, as at o-. For such a star the angle up v, expressing the lagging of the meridian, increases to a maximum for some situation of p between q and r, and decreases again to o at r; after which it takes an opposite direction, and tho meridian begins to get in advance of the star, and continues to get more and more so, till p has attained some point between * and P, where the advance is a maximum, and thence decreases again to o when j> has completed its circuit. For

s a

any star Bo situated, then, the mean of all the days so estimated through a whole period of the equinoxes is an absolute sidereal day, as if precession had no existence.

(911.) If we compare the sun with a star situated in the ecliptic, the sidereal year is the mean of all the intervals of its arrival at that star throughout indefinite ages, or (without fear of sensible error) throughout recorded history. Now, if we would calculate the synodic sidereal revolution of the sun and of a meridian of the earth by reference to a star so situated, according to the principles of Art. 418., we must proceed as follows: Let D be the length of the mean solar day (or synodic day in question) d the mean sidereal revolution of the meridian with reference to the same star, and y the sidereal year. Then the arcs described by the sun and the meridian

in the interval D will be respectively 360° — and 360° -r .

And since the latter of these exceeds the former by precisely 360°, we have

360° -j=360° -+360°;
d y

whence it follows that

■5 = 1 + 5 = 1-00273780,

y taking the value of the sidereal year y as given in Art. 383,

viz. 365d 6h 9m 9-6". But, as we have seen, d is not the absolute sidereal day, but exceeds it in the ratio 1 "00000011 : 1. Hence to get the value of the mean solar as expressed in absolute sidereal days, the number above set down must be increased in the same ratio, which brings it to 1*00273791, which is the ratio of the solar to the sidereal day actually in use among astronomers.

(912). It would be well for chronology if mankind would, or could have contented themselves with this one invariable, natural, and convenient standard in their reckoning of time. The ancient Egyptians did so, and by their adoption of an historical and official year of 365 days have afforded the only example of a practical chronology, free from all obscurity or complication. But the return of the seasons, on which

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