served; its consequences are much more easily conceived of, and it affects only a few of the binary and multiple stars whose motions have been watched. To distinguish the two kinds of aberration, the former has been called subjective, the latter objective aberration. Objective aberration then applies to those apparent displacements which originate in the length of time occupied in the transmission of light from the luminary to the eye.

§ 409. Let us consider a Binary Star, consisting of two stars, A and B, at so great a distance that light requires x years to reach the earth. Suppose, for simplicity, the common proper motion of the centre of gravity to be in a direction perpendicular to the visual ray. It is obvious that each of the two stars A and B will be seen, independent of the other at any given moment, not in the place which it occupies at that moment, but in that which it did occupy x years since, without regard to any change which may have taken place in its velocity or direction. Since this is true of each individual, it is true of both together, regarded as forming a compound luminary, the parts of which must have, with respect to each other and to the spectator, an apparent situation identical with their real situation x years ago. We see, therefore, the compound object A and B in the state in which it really did exist x years previously. This being true at every instant, it follows that in viewing such a system continually for a series of years, we necessarily perceive its orbit in its true form; and all the angles of position and distances in that orbit will be given truly by our measurements, unaf fected by any optical illusion or distortion whatever, only for an epoch antecedent by x years.

§ 410. There is a curious difference between the consequences of these two kinds of aberration, as shown in the heavens. Subjective aberration prevents our ever seeing a star at any moment in its proper place. We see it displaced by turns on every side. Objective aberration never removes a star from its true course, it merely causes it to lag behind and appear in a place historically, but not at the moment, true.

The sun, moon, and planets suffer both subjective and

objective aberration. Their subjective aberration may be found in the same way as that of the stars. Their objective aberration differs from that of the stars in this, that as we know their distances, we may ascertain how much they appear to lag behind; we may know how long since the place they now appear in was the true one.

The sun being always in the celestial ecliptic, and at the distance of 180° from the earth, he is always 90° before the point toward which the earth is moving; the aberration, therefore, in his case has always its greatest value, of 20."246. It always affects the longitude only, being in the plane of the ecliptic; and it always diminishes the apparent longitude, because it brings the sun nearer the point toward which the earth is moving. We always, therefore, by the effect of aberration, imagine the sun to be in a point 20."246 behind the true direction of the ray by which we see him.

The apparent places of the planets are affected in an analogous manner. The true place of any planet at the time of observation differs from the observed place by the arc which the planet describes in the time that a ray of light takes to pass from it to the earth. We must also allow for the motion of the earth during the same time.

§ 411. In these calculations we have considered the earth's motion in its orbit only, and have placed the observer in the line joining the earth's centre and the sun. There must however be some aberration produced by the earth's rotation. The greatest possible velocity from this cause is that of a point in the equator, .2916 of a mile a second. The greatest aberration from this cause will be to that arising from revolution in proportion to their respective velocities: 19: .2916: 20."246 .3108 of a second, a quantity too small to be regarded.

We have also supposed the earth's motion in her orbit to be circular and uniform. The variations from these causes do not exceed 0.003. It would therefore be an unnecessary refinement to allow for them.

CHAPTER VIII.

TIME.

Natural Divisions of Time. The Solar and Sidereal Day. Mean and Apparent Time. The Equation of Time. Variation in the Length of the Seasons. The Sidereal, Equinoctial and Anomalistic Years. Leap Year. Further Divisions of Time.

§ 412. By seeing things change, and have a beginning and end, we acquire the idea of before and after. By equal intervals of time we mean successions of events during which we can execute the same things in the same manner, or during which the same phenomena are reproduced in the same order. Unequal intervals are those in which we cannot do the same things, traverse the same road, or perform the same labor. It is then motion which gives the idea of time, and by motion time is measured; for we cannot trust our senses to measure either. Time and motion are the only measures of each other. We can describe a time only by mentioning what portion of her daily or yearly course the earth has performed while it lasts; we can describe the swiftness of a motion only by saying how much time it occupied. Thus the motion which we describe as occupying a given time must ultimately be referred to the time the earth occupies in another motion.

§ 413. As we have described the motion of a body by saying that it moves a certain distance in a certain time, so we may now divide such portions of time as we can grasp into definite periods during which certain motions last. We cannot conceive of the beginning or end of time, but we can form a definite idea of periods of time. The most obvious periods are those of the revolution and the rotation of the earth, a year, and a day. But we find more than one kind of day, and more than one kind of year. A day, in common language, is the period between two successive appearances of the sun on the meridian, or between his rising and setting. A year is the period dur

ing which all the seasons return; it may be reckoned from mid-winter to mid-winter, or from the vernal equinox to the vernal equinox, or from a point at a certain distance from those points to the same again. But these divisions which are so obvious, and which regulate the labors of the hus bandman and the common operations of life, are not nearly precise enough for astronomical purposes. The returns of the heavenly bodies to the meridian divide the year into convenient portions, the return of the sun to the same place in the heavens serves to mark when a larger portion of time has elapsed, but we must have some invariable standard of time, independent of the motions of the earth, with which to compare them. Such a standard we find in the oscillations of the pendulum. For the oscillations of a pendulum of a certain length, in a given latitude, must, in consequence of gravity, always occupy equal portions of time. Having this standard we may now ascertain whether these natural periods are always of the same length.

§ 414. The two natural days are the solar and the sidereal. Of these the solar day is most convenient for common use, because every one knows how often in the year, and when, the sun is on the meridian, and his presence there, and his rising and setting, control all our movements. But the sidereal day has this advantage, it is invariable.

As the orbit of the earth is but a point compared to the fixed stars, the revolution of the earth does not change the length of a sidereal day. A place on the earth which is under a certain star at noon one day, is again under it in 23h. 56m. 4"; and returns to it after the same interval throughout the year. The different portions of each day are also described in proportional periods, as is shown by the apparent motion of the stars. If two stars are in the same circle of rotation, but one is 180° distant from the other, half of a sidereal day elapses between their appulses to the meridian. If they are 90°, a quarter of a sidereal day elapses; and so on for every proportion of distance. Therefore not only is the duration of a sidereal day constant, but during every part of it rotation goes on uniformly. The pendulum also confirms these results.

§ 415. A solar day is longer than a sidereal day. For while the earth has been once rotating, she has advanced in her orbit nearly one degree, and this makes the sun appear to have advanced one degree. A place therefore which has a certain star and the sun on its meridian one day at noon, must turn round further to bring the sun again on its meridian than to bring the star. Thus an absolute turn of the earth on its axis falls short of a natural day, and the earth requires as much more than one turn on its axis as it has gone forward in that time, which on an average is part of a circle. Hence in 365 days the earth turns 366 times on its axis. Thus there is one more sidereal day in the year than there are solar days of the earth or of any other planet; one turn being lost by each planet's motion round the sun. From a similar cause the traveller who journeys eastward round the world loses one day let him take what time he will.

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§ 416. We find by the pendulum that the solar days are not equal in length one to another, nor are they at all seasons of the year uniformly described. Yet the sidereal day will not answer to regulate the employments of life. For as the sun moving continually eastward is in the course of the year at all distances east of the star chosen to mark the noon of a sidereal day, if we reckoned by the star we should in the course of the year have noon when the sun was rising or setting, or at midnight. The sidereal day is therefore useless for common life, although it is employed by astronomers.

We can, however, take the average of the solar days throughout the year, and thus obtain a mean solar day whose commencement never differs from that of the actual solar day much more than 16 minutes. The mean solar day is divided into 24 hours, each hour into 60 minutes, each minute into 60 seconds, and these are each of a fixed and determinate length. A pendulum 39.13929 inches, in the latitude of London, 51° 31' 1", in a vacuum at the level of the sea, vibrates seconds. The pendulum of an astronomical clock is usually made of such a length as to vibrate sidereal seconds. For the sidereal day may likewise be divided into hours, minutes, and seconds. Compu