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incidence, we have in this position of the earth summer in our northern hemisphere, whilst in the southern there is the contrary of all this, and consequently winter.

As the earth moves from position 1 to position 2, the excess of the day over the night diminishes, until at position 2, that is, on the 21st of March, it vanishes, and there is an equality, or it is the equinox. From 2 to 3, the night of the northern hemisphere continually gains on the day, and the difference is at 3, on the 21st of December, the greatest. From 3 to 4 it diminishes, and vanishes again at 4, which is the autumnal equinox.

VII.

SOLAR TIME.-SIDEREAL TIME.

WITH regard to the annual motion of the earth, one of the first things which it is important to remark is,—that whilst it is revolving from one point in its orbit to the same point again, it does not make a complete number of revolutions. Thus, if at the instant when the meridian of any place is passing over a fixed star, the precise place of the earth in its orbit be ascertained, then, at the instant when it has returned to the same place in its orbit, that meridian will not be passing over the same star as it would be if the earth in the interval had made a complete number of revolutions,--the revolution which it last commenced will remain uncompleted.

The number of complete revolutions which the earth will have made is 366; and of its 367th revolution it will have described that portion which it occupies 6' 9' 9:6" to describe. Now we usually say that there are 365 days and a quarter in a year, and each day is produced by a revolution of the earth upon its axis; how is it, then, that the number of revolutions is thus greater by one than the number of days ? This will readily be understood by referring to the figure, p. 315 of our last number. Let A, B, C represent successive positions of the earth in her orbit, s the sun, and AP, BP, cp lines drawn from the centre of the earth to a fixed star, which are to be considered parallel to one another, because of the distance of the star. Suppose the meridian of any place on the earth's surface to pass over this star, and the sun at the same instant in the position A. In the position B, the star will appear at N and the sun at K, and the meridian revolving in the direction K N B, will have to describe the angle nos, after passing over the star, before it can pass over the sun; but when it passes over the star it will have completed a certain number of revolutions exactly from the time it left the position a; that is, it will have completed a certain number of sidereal days; and when it passes over the sun it will have completed a certain number of solar days exactly: a certain number of revolutions, or sidereal days, is, therefore, completed before the like number of solar days, or alternations of day and night, is completed.

Now, the angle NBK, which the meridian has to describe after completing a certain number of sidereal days, before it completes the like number of solar days, is equal to the angle asb which the earth has, in the mean time, described about the sun; when, therefore, the earth has completed its revolution about the sun, or described 360°, the angle which the meridian has to describe, after completing a certain number of sidereal days, before it completes the same number of solar days, is 360°; but this 360°, being a complete revolution, will take it just another sidereal day to describe, which will make the whole number of sidereal days one more than the whole number of solar days.

VIII.

THE DIVISIONS OF TIME.-TRUE TIME.—Mean TIME.

This sum,

That division of time which is most obviously presented to us by the phenomena of the heavens, and which, as long as the world lasts, will continue to be the great practical division of time, is the alternation of a day and a night. This great division was established in the beginning of things, when God first divided the light from the darkness, and “the evening and the morning were the first day.” But a very slight observation is sufficient to show us, that the length of the period of light, and the length of the period of darkness, is perpetually varying,—that, for instance, the day of summer is longer than the day of winter, and that an opposite relation obtains with regard to the nights of these two seasons; but that the sum of these two periods, is, all the year round, and all the world over, nearly the same. which is nearly, and was at first imagined to be exactly, uniform, was called a day. Thus we understand the full force of the expression, “ the evening and the morning were the first day.” The first measurement of the length of the time of light and the length of the time of darkness, was no doubt made by observing the time between sunrise and sunset, and between sunset and sunrise; and this method admits of considerable accuracy. It would soon, however, be found to be at once more convenient and more accurate to observe the interval between two apparent passages of the sun over the meridian, or two of its greatest successive elevations in the heavens. Before instruments applicable to the exact admeasurement of angles came to be used, this was done by observing the interval between the times on two successive days when the length of the shadow of a vertical object was least.

This period is the true solar day. It is divided into 24 equal parts, called hours; and if these be counted from noon up to noon again, it is the astronomical day.

Now observations of so rough and uncertain a kind as those spoken of above, are yet sufficient to establish the fact that this solar day is not constantly of the same length. This irregularity of the length of the solar, as compared with the sidereal day, arises principally out of two

The first is, that the sun's apparent path round the earth is not parallel to the apparent paths of the stars,—or, in other words, that the axis about which the sun apparently revolves round the earth every year (the axis of the ecliptic,) does not coincide with the axis about which the earth revolves every day. The second cause is the continual variation of the motion of the earth in its elliptical orbit.

causes.

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 apps, 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.

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. The 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 PM 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, 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

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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, eyen 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 Nos 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. Ist. 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' 81"' every day, through which arc the meridian will revolve in 3 56" of sidereal time. If, therefore, p be the position of such a sun on one day, and m, at a distance 59' 8}" from it, be its position on the next, then will the meridian'nps arrive at m, 3' 564" 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' 561" short of one entire revolution, in the period of one entire revolution of the meridian, ás 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, P, (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 n p s 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 op mp is a right angle, p p is the hypothenuse of a right-angled triangle, and is therefore greater than op m. 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 pp such that op p=P m. 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.

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Thus, about the equinox the time of true noon precedes the time of mean noon, by reason of the excess of the space p p described in a mean solar day by the true sun, over that p 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

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then continually diminish until at the solstice p it vanishes; for it is manifest that there the corresponding ares of the ecliptic and equinoctial are equal; so that supposing, as we have done, that the true and mean

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 P, 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.

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