appears in the meridian at the same spot, it rises always in one point of the East, and sets in one point of the West; but all these things are different in the case of the sun. If we observe his height when he crosses the meridian, we find that it is less on the 21st day of December, than at any other period during the year; that it is greatest on the following 21st day of June; and that in the interval it is on each succeeding day greater than on the preceding one: and again, after the 21st of June the sun passes the meridian at points successively lower on each day, until, on the 21st of December, he again returns to his least elevation. The variation of this elevation is not uniform; it is most rapid about the 21st of March and the 21st of September; and for a few days before and after the 21st of June and the 21st of December it is hardly perceptible. At these periods therefore the sun has loosely been said to stand still; and they have in consequence, and from the seasons at which in Europe they happen, gained the name of the summer and winter solstices. The same observation may be made also with respect to the points where the sun rises and sets; indeed it is a necessary consequence of the sun rising from day to day higher on the meridian, that his points of rising and setting should also approach the pole which is above the horizon; and, on the other hand, when he is falling on the meridian, that the points of rising and setting should in like manner recede from it. It is also found by nice observation, that during the period in which his meridian heights are continually increasing, he always, when it is the North Pole which is above the horizon, sets nearer to the Northern point than he rises, and rises on the following day still nearer the North; and conversely, that when his meridian heights are continually diminishing, his point of setting on any day is more Southward than that of his rising that day, but less so than that of his rising on the following day. All these observations correspond exactly to the supposition, that for the six months extending from the winter to the summer solstice, the sun, besides partaking of the general rotation of the heavens, has a oner motion of his own in the heavens continually carries him Northand again, that in the interval be he summer and the winter solstice, also a proper motion, but that

its direction is opposite to the former, that is to say, from North to South. The same observations may be made where the South Pole is above the horizon, with similar results; the only difference being, that in that case his motion Southwards produces effects corresponding to those of motion Northwards in the other.

This however is not the only motion which we can discover in the sun. Every one who has observed the nightly appearances of the heavens with any attention, is aware that they continually differ. On each succeeding night, or it may be more convenient, for the sake of marking the changes more strongly, to take nights at a considerable interval, as for instance a month, from each other, some stars become visible which had not been so at the last time of observation, and others cease to appear which had then been seen. The new stars which from time to time make their first appearance, do so invariably in the Eastern portion of the heavens a little before sun-rise; those which have ceased to appear always were last seen in the West a little after sun-set. It is found also, that stars which rose a little before the sun at the former observations, rise longer before him at the latter; and, in the same manner, that those which se a certain time after him at the former set a less time after him at the latter observations. It is easily seen tha these changes can only be occasioned in one way. We have already seen tha the stars themselves keep the same po sitions with respect to each other; bu the sun evidently approaches the region of the heavens which are above, bu near his point of setting in the Wester part of the horizon, at the time of h setting; for having previously set befor stars there situated, he now sets at th same time with them. In the san manner, he recedes from the regions the heavens which are near, but abo his point of rising in the eastern part the horizon when he rises; for havi previously risen at the same time wi stars there situated, he now rises aft them. He evidently therefore has a m tion from West to East. These obs vations hold alike in every period of t year, and at the close of the year sun is in the same position relatively the stars, as at its beginning. We ha therefore ascertained that he has a p per motion from North to South, back from South to North, which resto him at the end of the year to the sa

altitude when on the meridian which he had at its beginning; and that he has also a motion from West to East, which in the same period brings him back to the same place, but which, being continually in one direction, can only do so by making him describe in that direction the complete circuit of the heavens.

SECTION 3.-Mode of ascertaining the motions and places of heavenly Bodies -Measure of time-Pendulum-Solar and sidereal day- Measure of place-Right ascension-Declination -Sun's path, the ecliptic - Equinoctial points-Longitude--Latitude. We are still however far from a sufficient knowledge of the motions either of the sun or stars. We see that they revolve round the earth, but that they do not do so in the same time, and we cannot yet tell whether either revolve uniformly, or with a variable motion. To know any thing certain concerning motion we must know the space passed over and the velocity with which it is so. But the velocity can only be measured by the space passed over in a given portion of time, and the first great object therefore is to find out some method of computing and measuring time. If we knew that the motion either of the sun or of the stars was equable, we might adopt that motion as a standard; but it is evident that we have as yet no reason for so supposing. An hour appears a period of a very different length according to the mode in which we pass it: almost imperceptible sometimes in sleep, short in agreeable mental or bodily exertion or amusement, long in irksome employment or tedious idleness; and we have no power, by mere sensation, of accurately discriminating or comparing the lengths of several unconnected periods. But the periods of the successive returns of a heavenly body, whether it be sun or star, to the meridian are unconnected; and we consequently have no means by mere mental perception of comparing their durations, and saying whether they are equal or unequal. And this being the case, we can have no right to assume either as a standard of time; for time, like everything else, can only be measured by reference to something of a fixed and determinate value. Our notions of time indeed are so complicated with the words describing the periods into which we find it convenient to divide it, that it requires some attention to feel

the full force of the difficulty. A day is the period to which we commonly refer everything, and which we consider as of a fixed and uniform duration; a day is also the period between the successive appearances of the sun on the meridian; and in common language it is also the period between his successive risings or settings. Being thus in the habit of using the word both as an expression for a certain fixed and uniform period of time, and for the interval between certain

events which do in fact happen very nearly at that distance of time from each other, we cannot readily separate the two meanings of the word, or per

Iceive how difficult it would be to discover the equality of the intervals between these events unless we had some certain and definite standard, independent of themselves, whereby to measure them.

The difficulty however certainly exists; and we may in some degree be convinced of its force, even by an argument drawn from the difference between the two latter uses of the word day which we have mentioned. During the period in which the sun's motion is from South to North, we have already seen that he rises farther North on each successive morning, and it necessarily follows (a consequence which will be fully proved hereafter), that on each successive day a larger portion of his circle of rotation is above the horizon. If his motion be such that each successive interval between his appearances on the meridian is equal, or nearly so, his successive appearances on the horizon will be rapidly accelerated for a considerable period, then slowly so, and finally retarded. This will be more fully explained hereafter; but we all know the fact from the common tables of the time of sun rise. Thus, in our climate, in March, sun-rise precedes the arrival of the sun on the meridian by a period increasing from day to day by about two minutes; in June the alteration is hardly perceptible; and in September the interval diminishes as rapidly as it increased in March. In March then, the day, understood as the interval from sun-rise to sun-rise, is about two minutes less than the day, understood as the interval from noon to noon; while in June they are equal, and in September again, the day from sun-rise to sun-rise is as much longer than that from noon to noon, as it was shorter in March. No man could pretend to be conscious of these minute differences of duration; and if there

.

appears in the meridian at the same spot, it rises always in one point of the East, and sets in one point of the West; but all these things are different in the case of the sun. If we observe his height when he crosses the meridian, we find that it is less on the 21st day of December, than at any other period during the year; that it is greatest on the following 21st day of June; and that in the interval it is on each succeeding day greater than on the preceding one and again, after the 21st of June the sun passes the meridian at points successively lower on each day, until, on the 21st of December, he again returns to his least elevation. The variation of this elevation is not uniform ; it is most rapid about the 21st of March and the 21st of September; and for a few days before and after the 21st of June and the 21st of December it is hardly perceptible. At these periods therefore the sun has loosely been said to stand still; and they have in consequence, and from the seasons at which in Europe they happen, gained the name of the summer and winter solstices. The same observation may be made also with respect to the points where the sun rises and sets; indeed it is a necessary consequence of the sun rising from day to day higher on the meridian, that his points of rising and setting should also approach the pole which is above the horizon; and, on the other hand, when he is falling on the meridian, that the points of rising and setting should in like manner recede from it. It is also found by nice observation, that during the period in which his meridian heights are continually increasing, he always, when it is the North Pole which is above the horizon, sets nearer to the Northern point than he rises, and rises on the following day still nearer the North; and conversely, that when his meridian heights are continually diminishing, his point of setting on any day is more Southward than that of his rising that day, but less so than that of his rising on the following day. All these observations correspond exactly to the supposition, that for the six months extending from the winter to the summer solstice, the sun, besides partaking of the general rotation of the heavens, has a proper motion of his own in the heavens which continually carries him Northward; and again, that in the interval be tween the summer and the winter solstice, he has also a proper motion, but that

its direction is opposite to the former, that is to say, from North to South. The same observations may be made where the South Pole is above the horizon, with similar results; the only difference being, that in that case his motion Southwards produces effects corresponding to those of motion Northwards in the other.

This however is not the only motion which we can discover in the sun. Every one who has observed the nightly appearances of the heavens with any attention, is aware that they continually differ. On each succeeding night, or it may be more convenient, for the sake of marking the changes more strongly, to take nights at a considerable interval, as for instance a month, from each other, some stars become visible which had not been so at the last time of observation, and others cease to appear which had then been seen. The new stars which from time to time make their first appearance, do so invariably in the Eastern portion of the heavens a little before sun-rise; those which have ceased to appear always were last seen in the West a little after sun-set. It is found also, that stars which rose a little before the sun at the former observations, rise longer before him at the latter; and, in the same manner, that those which set a certain time after him at the former, set a less time after him at the latter observations. It is easily seen that these changes can only be occasioned in one way. We have already seen that the stars themselves keep the same positions with respect to each other; but the sun evidently approaches the regions of the heavens which are above, but near his point of setting in the Western part of the horizon, at the time of his setting; for having previously set before stars there situated, he now sets at the same time with them. In the same manner, he recedes from the regions of the heavens which are near, but above his point of rising in the eastern part of the horizon when he rises; for having previously risen at the same time with stars there situated, he now rises after them. He evidently therefore has a motion from West to East. These observations hold alike in every period of the year, and at the close of the year the sun is in the same position relatively to the stars, as at its beginning. We have therefore ascertained that he has a proper motion from North to South, and back from South to North, which restores him at the end of the year to the same

altitude when on the meridian which he had at its beginning; and that he has also a motion from West to East, which in the same period brings him back to the same place, but which, being continually in one direction, can only do so by making him describe in that direction the complete circuit of the heavens.

SECTION 3.-Mode of ascertaining the motions and places of heavenly Bodies -Measure of time-Pendulum-Solar and sidereal day- Measure of place-Right ascension-Declination -Sun's path, the ecliptic - Equinoctial points-Longitude-Latitude. We are still however far from a sufficient knowledge of the motions either of the sun or stars. We see that they revolve round the earth, but that they do not do so in the same time, and we cannot yet tell whether either revolve uniformly, or with a variable motion. To know any thing certain concerning motion we must know the space passed over and the velocity with which it is so. But the velocity can only be measured by the space passed over in a given portion of time, and the first great object therefore is to find out some method of computing and measuring time. If we knew that the motion either of the sun or of the stars was equable, we might adopt that motion as a standard; but it is evident that we have as yet no reason for so supposing. An hour appears a period of a very different length according to the mode in which we pass it: almost imperceptible sometimes in sleep, short in agreeable mental or bodily exertion or amusement, long in irksome employment or tedious idleness; and we have no power, by mere sensation, of accurately discriminating or comparing the lengths of several unconnected periods. But the periods of the successive returns of a heavenly body, whether it be sun or star, to the meridian are unconnected; and we consequently have no means by mere mental perception of comparing their durations, and saying whether they are equal or unequal. And this being the case, we can have no right to assume either as a standard of time; for time, like everything else, can only be measured by reference to something of a fixed and determinate value. Our notions of time indeed are so complicated with the words describing the periods into which we find it convenient to divide it, that it requires some attention to feel

the full force of the difficulty. A day is the period to which we commonly refer everything, and which we consider as of a fixed and uniform duration; a day is also the period between the successive appearances of the sun on the meridian; and in common language it is also the period between his successive risings or settings. Being thus in the habit of using the word both as an expression for a certain fixed and uniform period of time, and for the interval between certain events which do in fact happen very nearly at that distance of time from each other, we cannot readily separate the two meanings of the word, or perceive how difficult it would be to discover the equality of the intervals between these events unless we had some certain and definite standard, independent of themselves, whereby to measure them.

The difficulty however certainly exists; and we may in some degree be convinced of its force, even by an argument drawn from the difference between the two latter uses of the word day which we have mentioned. During the period in which the sun's motion is from South to North, we have already seen that he rises farther North on each successive morning, and it necessarily follows (a consequence which will be fully proved hereafter), that on each successive day a larger portion of his circle of rotation is above the horizon. If his motion be such that each successive interval between his appearances on the meridian is equal, or nearly so, his successive appearances on the horizon will be rapidly accelerated for a considerable period, then slowly so, and finally retarded. This will be more fully explained hereafter; but we all know the fact from the common tables of the time of sun rise. Thus, in our climate, in March, sun-rise precedes the arrival of the sun on the meridian by a period increasing from day to day by about two minutes; in June the alteration is hardly perceptible; and in September the interval diminishes as rapidly as it increased in March. In March then, the day, understood as the interval from sun-rise to sun-rise, is about two minutes less than the day, understood as the interval from noon to noon; while in June they are equal, and in September again, the day from sun-rise to sun-rise is as much longer than that from noon to noon, as it was shorter in March. No man could pretend to be conscious of these minute differences of duration; and if there

fore we were to depend on sensation merely for our measure of time, we might just as well take the interval from sun-rise to sun-rise for our standard, as that from noon to noon; and either as that from the successive appearances of a given star on the meridian. The differences between the actual lengths of these different periods may seem trifling; but such is the nicety required in astronomical observations and computations, that no inaccuracy can safely be overlooked; nor, if we once force ourselves to give up the appearance of security given by the popular use of the words " day," "year," &c., are we justified in believing that our errors will be confined within any narrow limits.

We have however one standard of time independent of any assumption whatever, except that of the truth of those general principles, called the laws of motion, which are involved in all the deductions of mechanical science, and which, although inferred from observation and reasoning, do not admit of strict and absolute demonstration. All bodies are found, by experiments made anywhere on the earth's surface, to fall there towards the earth in a straight line, called a vertical line, in consequence of the action of some force acting on them in that direction, which we call the force of gravitation. If at that point the force act at every instant of time with the same intensity, a pendulum properly adjusted will perform its successive oscillations in equal times; that is to say, it will always take the same time in swinging backwards and forwards. This depends on no assumption whatever; it is a necessary mathematical consequence of the action of a constant force, tending vertically downwards, on such a body so suspended. If therefore we can ascertain that the force by which a body falls to the ground at a particular place is always equal, we know that the oscillation of a pendulum there will give a fixed standard of time. We might perhaps venture at once to assume that this force is invariable, on the principle that, all the circumstances under which it acts being, as far as we can perceive, the same at all times, there is no reason whatever for supposing it to vary. It is, however, satisfactory, if possible, to ascertain by actual experiment, whether the force be variable or not. Of course, when the object is to ascertain the mode of measuring time, no measure

of time must be used in the process. Nothing, therefore, which depends upon the measurement of velocity can be admitted, for velocity only means space described in a certain time. We have however sufficient, though not very ready means of ascertaining the fact. Space is easily measured; and the momentum of a body can be, although not very accurately, yet sufficiently estimated by the effects it produces. If the quantity of matter in a body be represented by A, and the force acting on each particle of matter in it by F, and the space through which it falls, by S, it may be proved, as a necessary consequence of the meaning of these terms, that the momentum varies as A. F., S.** If therefore we try the experiment with the same body falling through the same space at different times, the momentum will vary as F, and will increase and diminish with the increase and diminution of the force. If then we find that the body always sinks to the same depth in similar bodies placed to receive it, that it produces the same contraction of a spring attached to the point on which it falls, and generally that it produces effects always equal in manners which may be varied according to the pleasure of the experimenter, we are led to the conclusion that the force always acts alike at the same point. Now this is the conclusion found by experiment.

We have therefore in

the pendulum a standard of duration: not a convenient one as yet, because we have nothing whereby to fix on any particular length of pendulum as the standard; nor a general one, for we have as yet no reason to form an opinion whether the force of gravity at different points on the earth's surface is the same or different, and we therefore cannot tell whether our pendulum of one place would swing at the same rate at another. We know only that at the place where the experiments were tried its successive oscillations are always of equal lengths, and consequently, that we may there ascertain by it whether the revolutions of the sun, or stars, or either of them, be of uniform length; for whatever be the duration of the uniform oscillation