Let NCSL represent the sphere of the heavens, EQ the equinoctial, CL the ecliptic, N, s the poles, p the place of the sun in the ecliptic, at the time when the celestial meridian NPS of any place is passing over it, m the place of the sun in the ecliptic on the following day. To pass over the sun on this following day, the meridian, after completing its revolution into the position NPPs, must further revolve through the angle PNM into

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the position NMS; now the meridian revolves uniformly; if, therefore, the angle through which, in order to overtake the sun, it has to revolve every day, over and above a complete revolution, be the same, then will the length of time between its leaving the sun and returning to the sun again be the same,-or, in other words, the solar day will be always of the same length; but, on the contrary, if this angle be not always the same, the lengths of successive solar days will, for this cause, be different. Now, supposing the sun to move uniformly in the ecliptic, it is manifest that this angle cannot always be the same, because the ecliptic is oblique to the equator.

The

It is manifest that as the meridian revolves uniformly, it would carry a point fixed upon it uniformly; and if such a point were fixed upon it, half way between the poles, it would carry it along the equinoctial. meridian traverses, therefore, the equinoctial uniformly, and equal spaces on the equinoctial are revolved over in equal times by the meridian, or correspond to equal angles described by the meridian; if, therefore, equal spaces on the ecliptic corresponded to equal spaces on the equinoctial,that is, if taking distances anywhere on the ecliptic, each equal to one another, and to pm, the spaces P M on the equinoctial corresponding to them were all of necessity equal to one another, then the corresponding angles PNM would all be equal; and if pm were the space described by the sun in the ecliptic every day in the year, then would every solar day be of the same length. But this is not the case. If equal spaces, such as pm, be taken on different points of the ecliptic, it will be found, and it is manifest, that the spaces such as PM, corresponding to these on the

equinoctial, are not equal,-the angles PNM corresponding to equal motions of the sun in the ecliptic are, therefore, not equal; and the solar day is not then, at all periods of the year, of the same length, and would not be, even if the sun's motion in the ecliptic were regular. But the sun's motion in the ecliptic is not regular, because the earth's motion in its orbit is not regular. Referring to the fig. in page 315 of the last number, we perceive, that the angle NBS being equal to the angle ASB, if the latter angle, representing the earth's angular motion in any given time about the sun, be not always the same, then the angle NBS or the arc N K, representing the sun's apparent angular motion in the ecliptic in that time, will not always be the same. Now we know, and it will be shown hereafter, that the earth's angular motion about the sun is varied, because its distance from the sun varies continually. Thus, then, the irregular motion of the sun in the ecliptic is accounted for; it sometimes describes 57′ of the ecliptic in a day, and sometimes 61'; and from this cause arises a difference in the length of the solar day which may amount to 8′ 20′′ of time. We have, then, two principal causes of irregularity, in the length of the solar day, and the true time of noon. 1st. The inequality of the angles through which the meridian must revolve on successive days to overtake the sun, caused by the obliquity of his path. 2dly. The irregularity of his motion in his path, resulting from the elliptic form of the earth's orbit. If we imagine a sun to traverse the equinoctial instead of the ecliptic, with a continued uniform motion in the period of each year, or in 365-2,422,414 days, it will describe an are of 59′ 8′′ every day, through which are the meridian will revolve in 3'56" of sidereal time. If, therefore, P be the position of such a sun on one day, and м, at a distance 59′ 8′′ from it, be its position on the next, then will the meridian NPs arrive at м, 3′ 56′′ after completing one entire revolution of the heavens. If, therefore, we take a pendulum clock, and so regulate the length of its pendulum, that its hour-hand shall have completed 3′ 56′′ short of one entire revolution, in the period of one entire revolution of the meridian, as marked by two passages of the meridian over the same stars, then, 3′ 56′′ after this the meridian will pass over our imaginary sun, and, at the same instant, the hand of the clock will have completed its revolution. A clock thus regulated is said to be regulated to mean solar time.

Now let us suppose that our imaginary sun sets out from the point Aries,, (see the figure on the next page), at the instant of the vernal equinox, when the true sun is also in that point. Let the dial-plate of the clock be divided into 24 equal parts, and let the hand at that instant stand at 24. Also let the meridian NYS be at that instant passing over the sun. Let m be the position of the imaginary sun at the instant when the hand of the clock next points to 24, and the meridian is again passing over the imaginary sun. Since the angle mp is a right angle, ∞ p is the hypothenuse of a right-angled triangle, and is therefore greater than om. The true sun having, therefore, described in the ecliptic a space equal to that of the imaginary sun in the equinoctial, will be at some point p' in Yp such that op pm. Thus, then, when the meridian passes over m, it will have first passed over p', or it will have passed over the true sun before it passes over the imaginary sun, or before the hand of the clock again shows 24 hours.

Thus, about the equinox the time of true noon precedes the time of mean noon, by reason of the excess of the space p described in a mean solar day by the true sun, over that m described by mean sun. For some weeks the time by which the true thus precedes the mean noon will continue to increase, until it has attained an interval of about 16'; it will

then continually diminish until at the solstice P it vanishes; for it is manifest that there the corresponding arcs of the ecliptic and equinoctial are equal; so that supposing, as we have done, that the true and mean sun move each with the same uniform velocity, the meridian will pass over them both at the same time. True and mean noon coincide therefore at the solstices. After the solstice is passed, mean noon will begin to precede true noon, and the interval will again increase up to a certain point between this solstice and the following equinox; having then attained its maximum, it will begin to diminish, until at the equinox it vanishes, and mean and true noon again coincide. In passing on further to the next solstice, the time of true will begin to precede that of mean noon, and the same changes will be gone through as in the preceding half of the ecliptic, until both suns again come together, and both noons coincide in the point Aries, whence they set out. Thus, then, on the supposition which we have made, that the sun moves uniformly in the ecliptic, it appears that the time of true and mean noon will alternately precede one another, and that four times a year the interval between them will attain a maximum value*.

The sun does not, however, move uniformly in the ecliptic, by reason of the ellipticity of the orbit of the earth; and, moreover, the velocity of his apparent motion is dependent, not upon his position with respect to the solstitial or equinoctial points, but upon the position of the earth with respect to the principal points of her orbit about him, her

* This maximum value will be attained when the sun is 46° 14' from either equinox, and it may amount to 10' 3.9" of time.

aphelion and perihelion, the nearest and most distant points. Thus, then, the amount of the deviation of the motion of the sun, at any point of the ecliptic, from his mean motion, is dependent on the position of the perihelion of the earth's orbit in the ecliptic; moreover, this position is varying from year to year. Here, then, is another and most important cause of the variation of the time of true from that of mean noon, by reason of which cause alone it may be calculated, that at certain periods of the year the time of true noon would differ from that of mean noon by about 8' 20" of time.

It has been before stated that, by reason of the obliquity of the ecliptic alone, the times of true and mean noon might be made to differ 10′ 3·9′′ of time. If, then, the time of greatest variation from the one cause coincided with the time when the greatest variation takes place by reason of the other cause, then both thus conspiring, the whole variation of the time of true from that of mean noon would be not less than 18'23.9'. But this is not the case,—and the maximum interval between the time of noon as shown by a good clock keeping mean time, and the true time of the sun's passing the meridian, never exceeds 16′ 17′′ of time. Moreover, by reason of the irregularity introduced by the elliptic motion of the earth, the coincidence of the true and mean noon at the equinoxes and solstices is destroyed, and true noon is shown by the clock, not at the periods of the equinoxes and solstices, but at the following periods,-the 15th April, the 15th June, 1st September, 24th December. The sun gains upon the clock between the two first of these periods, loses during the second, and gains again during the third. It is behind the clock by its greatest interval of 14′37′′ on the 11th of February, and before it by its greatest interval of 16′17′′ on the 3d of November.

There are three methods of measuring time, commonly in use among

astronomers.

1. It is measured by sidereal time, which is regular, being governed by the regular revolutions of the earth upon its axis, as shown by successive returns of the meridian to the same star.

2. It is measured by mean time, the nature of which has been sufficiently explained in the preceding pages, and the method of regulating an astronomical clock, so as to show that time (see p. 358); this time, like sidereal time, is uniform, being dependent upon the period required by the earth to make one complete revolution in her orbit.

3. Solar, or true time, as it is called, which is measured by the time between two successive noons, or actual passages of the meridian of any place over the sun; and that time not being the same at all seasons of the year, it follows that solar time is irregular, and that the solar hour, which is the 24th part of the solar day, has not exactly the same length on any two successive days.

The difference between true and mean solar time, explained in the preceding pages, is called the equation of time. Clocks, called equationclocks, have been so constructed, that whilst one of their hands shows on the dial-plate mean time, the other points to true time. The mechanism of a clock, whose hand is to follow the irregular course of the sun, through each quarter of the year, is, however, so complicated, that little dependence can be placed upon it..

The sidereal day, which, like the solar, is divided into 24 hours, commences at the instant when the meridian, at the place of observation, passes over the equinoctial point Aries, and terminates when it returns to that point.

Thus the time of the sidereal day, when the meridian passes over any particular star, is the time which it takes to revolve from the equinoctial point Aries to that star; and since it revolves regularly through 15° in every sidereal hour*, it is manifest that, allowing at the rate of 15° for each sidereal hour shown by the clock, (or, as it is called, converting the time into degrees,) we may ascertain at once the right ascension of the star, which is no other than the number of such degrees intervening between it and the point Aries, by observing the sidereal time when the meridian passes over it.

Since the right ascensions of all the principal stars have been accurately ascertained; by observing the sidereal time shown by the clock when the meridian passes over any such star, we may conversely ascertain whether the clock be right or not; and it is thus that astronomical clocks are regulated.

Solar time is found by observing the time of two successive passages of the sun over the meridian, and dividing the interval into 24 hours. It is the time shown also by a well-constructed sun-dial.

Mean time is found by observing the true time, and allowing, according to the table of equation of time, for its difference from true time. Thus, to determine the mean time of noon, we should observe by our clock the true time of noon, or the exact time of the meridian passing over the centre of the sun. If then we deduct from this, or add to it, the equation of time for the noon of that day, the result will bring us to mean noon.

There is yet another, and practically a better method. If a clock be set to true mean time, the stars will every day complete an apparent revolution, that is, the meridian itself will complete a real revolution,precisely 3' 55.9" before the hour-hand has completed its revolution of 24 hours on the dial. Observe, then, two successive transits of a star; at the first set the hour-hand at 12, and regulate it so that at the second it shall show 3' 55.9" short of 12. It will then be regulated so as to show mean time. It only remains to set it at the mean noon, as explained in the preceding page.

* This is evident from the fact that it completes its revolution of 360° in 24 sidereal hours.