of our pendulum, if we observe that either of these periods of revolution always includes the same number of these oscillations, that is an uniform period, and may itself be adopted as a standard of duration. If the number of oscillations be different in different revolutions, the period is not uniform or fit for a standard*.

Now the period of revolution of the stars is found by this sort of comparison to be always accurately the same. Each star is found to have precisely the same interval between its successive appearances on, or as they are also termed, appulses to, the meridian, and this interval is called a sidereal day. It is also found that the different portions of each revolution are described in proportional periods: thus, if two different stars are in the same circle of rotation, but the one distant 180° from the other, the half of a sidereal day elapses between their appulses to the meridian; if the distance of two such stars be 90°, the interval between their appulses is a quarter of a sidereal day; and so in like manner for every proportion of distance. We find therefore, not only that the duration of a sidereal day is constant, but that during every part of it the rotation goes on uniformly; or in other words,

It must not be understood that the considera

a

tions stated in the text were those from which the expediency of using the pendulum as standard of time was in fact deduced, though they seem to furnish the mode of deducing it, whichinvolves the fewest assumptions. It is, indeed, probable that the experiments mentioned in the text have never been made with any degree of minuteness. Huygens, who first demonstrated the mathematical principles of the motion of a pendulum, though Galileo had accidentally observed that its oscillations appeared to be all of the same length, seems to have proceeded on the principle mentioned in the text, of assuming that the force of gravitation was constant, from the absence of any apparent reason for its varying. From this assumption he deduced the laws of the motion of a pendulum; and it has ever since been adopted as the standard of time.

+ These observations of distance cannot be made with any great degree of accuracy, unless by means, hereafter pointed out, which proceed on the supposition that the heavens do revolve continually with an uniform velocity. They may, however, be made with certainty enough to convince us that the limits within which any variation must be confined must be exceedingly small: for, by multiplying observations, and taking a mean, or average, between their results, we may be sure, where there is no cause affecting all the observations in the same way, to obtain a value very near the truth. This is an obvious and necessary consequence from the supposition that there is no constant cause of error affecting all the observations alike: any accidental cause is likely to affect them in different manners, sometimes increasing and sometimes diminishing

the apparent distance, and the errors thus compensating each other, the mean result, even where the individual errors are considerable, will not differ much from the truth.

that the heavens revolve round their axis continually with an. uniform velocity. The discoveries of an advanced state of science furnish the most complete confirmation of this result, by shewing that none of the causes which produce variations and disturbance in other motions, produce any in that from which the apparent motion of the heavens round an axis proceeds.

The interval between the successive appearances of the sun upon the meridian, or from noon to noon, is necessarily longer than that between those of a star; for as the motion of the heavens is from East to West, and the proper motion of the sun from West to East, the sun on each successive day, when the point of the heavens where he was at noon on the day before returns to the meridian, is to the Eastward of that point, and, consequently, to the Eastward of the meridian; and he therefore only returns to the meridian after the rotation of the heavens has continued for some additional period, long enough to bring his new place to the meridian. The interval between two successive appulses of the sun to the meridian, is called a solar day. We find by observations with our pendulum, that the length of the solar day is continually varying, but still that its variations succeed each other in a regular succession, and go through all their changes in a certain period of time, a year. Although therefore the solar day is of variable length, we can, as we know all its variations and their period, ascertain its mean or average length; and this quantity is called a mean solar day.

The pendulum, although it furnishes us with the means of ascertaining that the motions of the stars are uniform, and those of the sun variable, is not, for the reasons we have given, adapted in the first instance for furnishing the common standard of time. The solar day, again, does not seem well suited to the purpose, on account of its variable length. On the other hand, as the continual appearances and re-appearances of the sun above the horizon furnish the most remarkable distinctions between different portions of time, and do practically regulate the occupations of men, and determine the periods of labour and those of rest, it would be inconvenient so to fix the standard of time used respecting the common occurrences of life, that the periods by which we measure it should continually have

their commencement at different parts of this, which we may call the working day. This inconvenience however would clearly occur if we took the sidereal day for our ordinary standard; that is, if we fixed the commencement of the day, at the instant when a particular star is on the meridian: for as the sun, moving continually Eastward, is successively at all distances East of the star, he must come to the meridian at all intervals of time after it; and consequently the star will be on the meridian at one period of the year when the sun is so also, or as we say at noon; at others when he is just rising, just setting, or midway below the horizon, or as we say at midnight. The mean solar day, however, is liable to no such objection: being a period deduced from computation of the average of the actual solar days, it is a fixed and invariable period; and as no solar day differs much from it, and the differences are some in excess, and others in defect, the period of the observed commencement of the real solar day never differs so far from the computed or registered commencement of the mean solar day as to occasion the inconvenience which must result when these standards materially differ. In fact, the difference between the commencement of the real and the mean solar day never much exceeds sixteen minutes*.

Taking then the mean solar day as our standard of time, it is divided into twenty-four hours, each hour into sixty minutes, and each minute into sixty seconds; and these are each of fixed and determinate length. It is established by the principles of mechanics, that we can, by varying the length of a pendulum, make its oscillations of any exact length that we please: and as the second is the smallest division of time in common use, it is usual to make the pendulum of a clock of such a length that its oscillations are of a second each. If the force of gravity is different at different places on the earth (as we shall hereafter see that it is), the lengths of the pendulums vibrating seconds at these different places will differ; but they may be so adjusted that the period

Although the commencement of the mean solar day seems the most convenient division of time, for the reasons given in the text, it is by no means universally adopted. The French use the true solar day; that is to say, they fix the commencement of each day at the instant when the sun is

actually on the meridian. The length, therefore,

of the day varies, and their clocks require to be regulated accordingly.

of the oscillation shall be accurately the same in all. The length of a pendulum vibrating seconds in the latitude of London, 51° 31.'1, in a vacuum at the level of the sea, is 39,13929 inches.

Taking these common divisions of time into hours, minutes, and seconds, the length of the sidereal day is found to be uniformly 23 hours, 56 minutes, or more accurately, 23h 56m 4.092. This period may be divided into 24 equal portions, and each of those subdivided into 60, and those again into 60 other equal portions; and these divisions will be respectively sidereal hours, minutes, and seconds, bearing the same relation to the sidereal day that the common hour, minute, and second do to the mean solar day. Of course a pendulum may be made of such a length that its oscillations will be of a sidereal second each; and in fact astronomical clocks are usually made so. Computations and observations made by sidereal time are of course easily transferred to common (or mean solar) time, and the reverse.

Having now got our measures of time, we proceed to see how, by means of them, we can ascertain the actual position of the sun in the heavens on each particular day. His light so far overpowers that of every other body, that there is some difficulty in observing his situation accurately, by immediate comparison with that of others; for, although we can see the stars through telescopes in the day time, they are not very readily found, and cannot easily be observed unless we have some previous knowledge of their positions, and adjust the telescope accordingly; they conse quently are ill suited for the earliest observations. The moon indeed is often visible at the same time with the sun, and may be compared in her position with him by day, and afterwards with known stars by night; but she has, as we shall see, a very compli cated proper motion of her own, in the interval between the two observations; and consequently furnishes only an inconvenient method of determining the position of another body. We must therefore have recourse to other means, and for this purpose we must explain a few of the simpler properties of the sphere.

In fig. 2. let Pp represent the axis of the heavens, P, p, being the poles, and let PEPQ be the meridian of the observer, Suppose also that a plane is

drawn through O, the centre of the sphere, so that Pp shall be perpendicular to it; the intersection of this plane with the sphere will be a great circle of the sphere, and this great circle will bisect the meridian PEP Q, and every other circle drawn through the two points P, p, as PA, p, PA,p, PA, p. This circle EA, A, Q is called the equator. The meridian of every place is a great circle passing through the two poles, and every circle so passing must be the meridian of some particular places, according to our original definition of a meridian. Generally therefore they are called meridians, or meridional lines, and the equator bisects all the meridians. In this sense of the word the meridians are lines in the sphere partaking of its general revolution. The meridian of a particular place, on the contrary, is a fixed

Any circle formed by the intersection with the sphere of a plane passing through the centre of the sphere, is called a great circle of the sphere. If a diameter of the sphere be drawn perpendicular to this plane, the extremities of this diameter are called the poles of the great circle; and if any number of great circles be drawn through these poles, they will each be in planes perpendicular to great circle. Being in planes perpendicular to it, they are so, of course, to all other circles parallel to it. Thus, taking the circles whose names are given in the text, the north and south poles of the heavens are the poles of the equator, and every meridian is a secondary to the equator, and perpendicular to it, and to every circle parallel to it: or every meridian is perpendicular to every circle of daily rotation, for all such circles are parallel, and the equator is one of them.'

the first circle, and are called secondaries to that

It may also be convenient here to mention that all great circles in the same sphere are equal, and that they all bisect each other.

On all matters relating to the properties of the sphere, see Treatise on Geom. Book VI,

line, with which each of the others in the course of its revolution coincides. Thus, referring to the common celestial globe for illustration, lines drawn from pole to pole on the surface of the globe would be meridians, using that term generally: the brass meridian corresponds to the meridian of the place, for which the globe is adjusted.

The equator EA Q being a circle bisecting every meridian, has every point equidistant from P, or p: it is therefore a circle of rotation, or any point situate in the equator describes a course round the earth coincident with the equator itself. Let ea, aq be any small circle parallel to the equator. It will, by the properties of the sphere, have every point equidistant from P or p, and will consequently be a circle of rotation. If e, a1, az, da, be the several points where the different meridians Pe Ep, Pa, A, p, Pа, A,р, Pаs As p, cut this small circle, and E, A, Ag, A3, the points where the same meridians cut the equator, then it is a property of the sphere, that whatever be the proportions that the several arcs E A1, E A2, E Ag, bear to the whole circle EA, Q, the arcs e a1, e az, e as, bear the same proportions respectively to the whole circle ea, q. But the periods of rotation of a star in the small circle ea, q, or in the great circle E A, Q, are equal; for they are each a sidereal day; and the motions of rotation in each are uniform: the periods of rotation therefore through the arcs a, e, age, as e, are to the whole sidereal day, in the proportions of the respective arcs themselves to the

whole circle e aq; and the periods of rotation through the arcs A, E, A, E, A, E, are to the same whole sidereal day in the proportions also of those respective arcs to the whole circle E A, Q, the same proportion as the preceding ones. The periods of rotation therefore through the arcs a, e, A, E, are equal; and so are those through age, and AE, and again through a, e, and A, E. But when any point of one of these meridians coincides with the meridian of the place, the whole meridian coincides with it; or every point of a meridian is on the meridian of the place at the same time; and the periods of rotation through corresponding arcs being equal, every point of a meridian, when not on the meridian of the place, requires the same time to arrive at it; and that time bears the same proportion to a sidereal day that the arc of the equator, intercepted between the meridian of the place and the meridian in question, bears to the whole circumference of the equator. And conversely, if we observe the time when one star is on the meridian of the place, and that when another body appears there on the same side of the pole (for this qualification is necessary in the case of circumpolar stars which appear on the meridian of the place both above and below the pole), and note their interval, we know the interval of the equator intercepted between the two meridians; for it is the same proportion of the whole equator which the observed interval of time is of the whole sidereal day. It is of course convenient to have some fixed point of the equator as a standard of reference, and that point, which we shall presently see the reason of choosing, is called the first point of Aries, and is often marked thus. The distance, measured Eastward, from to the intersection with the equator of the meridian passing through any heavenly body, is called the right ascension of the body, and is obviously the same for every body upon the same meridian, and upon the same side of the pole, and for none off it. The right ascension may evidently be estimated in time, as it is ascertained by it. The period of a whole revolution, or of passing through 360°, is twenty-four hours of sidereal time: in one such hour therefore a body passes through 15° of right ascension; in four minutes through I°; and these right ascensions may be equally well called either 1° and 15°, or four minutes, and hour. In other words, an astronomical

one

clock marks 0h 0m when the first point of Aries is on the meridian; when it marks 4m the bodies then on the meridian are said to have 1° or 4m of right ascension: when it marks 1", the bodies on the meridian are said to have 15° or 1 of right ascension.

By the observation of time therefore we can ascertain the right ascension of a heavenly body, and consequently the meridian on which it is. If we can also observe its distance from the pole, or its distance from the equator, which is called its declination, and is called North or South declination as the body is North or South of the equator, we ascertain its position upon that meridian. The declination being measured along a meridian, the meridians have also the name of circles of declination. Every point in a small circle parallel to the equator, is of course at the same distance from the equator, or, in other words, has the same declination: such a small circle is therefore called a parallel of declination.

The observation of declination is easily made. The horizon of any given place is a fixed circle: the point therefore where a perpendicular to the horizon meets the heavens above the spectator's head (which is called the zenith*) is a fixed point also. This point may always be ascertained, for a plumb line hangs directly in a line from it. The zenith is obviously a point in the meridian; for the plane of themeridian passes through the place of observation perpendicular to the horizon, and the perpendicular to the horizon at the place must be a line in that plane.

The zenith is the pole of the horizon, and if any great circles be drawn through the zenith, they will be secondaries to the horizon. These are called vertical circles: the meridian of the place is of course one of them. The arch of a vertical circle intercepted between a heavenly body and the horizon, is called the altitude of the body, and can always be measured by proper instruments t.

The opposite point, where the perpendicular produced below the horizon meets the opposite hemisphere of the heavens, is called the nadir. are derived from the Arabian observers.

These, like many others of the terms of astronomy,

The observer being placed apparently in the centre of the sphere, the angle which two objects in the heavens subtend at his eye will accurately

measure the arc of the great circle of the heavens which lies between them. This arc therefore, or their distance, is accurately measured by obser

vation of the angle between them; or rather, as

we know nothing of their actual distances, we can only observe the angles, which give us the rela

tive directions of the bodies; and then we suppose

The distance from the body to the zenith, or its zenith distance, together with the altitude, make up the whole distance of the zenith from the horizon, or 90°. The zenith distance, therefore, is the complement of the altitude.

The pole is a fixed point in the meridian; it is therefore at a fixed distance from the zenith of any given place, and from the North and South points of the horizon there. If therefore, when any heavenly body is on the meridian, its distance from either the zenith or the North or South point of the horizon can be observed, its distance from the pole, or its North Polar distance, can be ascertained from it. And it is obvious that if an instrument be adjusted so as to move in the plane of the meridian, these distances may be ascertained by mere observations of altitude made with it: for the altitude of a body when on the meridian, is its distance either from the North or South point of the horizon; and the complement of the altitude is the zenith distance. The adjustment of the instrument to move in the plane of the meridian, is more complicated than that required merely to make it move in a vertical circle: it is however generally made with the best instruments, which are fixed so as to move in the plane of the meridian, and in that only. Having an instrument so adjusted, the north polar distances of all the stars may be ascertained.

Let S1, S2, S3, S4, in fig. 2, represent any heavenly bodies on the meridian: S, E, S, E, S, E, S. Q, are obviously their declinations. Now, observing that PE, or PQ 90°, it is obvious that whenever the North polar distance, as PS2, PS3, PS4, is less than 90°, the declination S, E, S, E, S, Q = 90° - North polar distance; and in these cases the

them ranged in the surface of a sphere, for the convenience of computation, by applying to them

the principles of spherical geometry.

The altitude of a body is easily ascertained. The

instrument with which the observation is made can be mechanically adjusted, so as to move in a

vertical plane, and, consequently, in the plane of a vertical circle. The place of the horizon may be ascertained, though some nicety is required in the ascertainment, into the details of which it is not necessary here to enter. All instruments intended for observation have a graduated circular limb attached to them, and when the plane of the horizon is ascertained, the points on the limb which are in that plane may be found; and the angle between the diameter joining them and the direction of the instrument, which obviously measures the altitude of the body, observed by means of the gra duation on the limb. There are additional and more delicate adjustments, which need not be detailed here.

declination is North. When the North polar distance, as PS1, is greater than 90°, the declination S, E = North polar distance - 90°. In each case therefore the declination is the difference between the North polar distance and 90°, being North when the North polar distance is less, and South, when it is greater, than 90°.

We can then always ascertain the declination of a body by observing its altitude when on the meridian of the place. Its right ascension may be ascertained in the manner already pointed out: or, if the right ascension of any particular star has been previously accurately ascertained by a sufficient number of observations, then the right ascension of any other star may be found by observing the time which elapses between its appearance on the meridian and that of the star, whose place we suppose to be already determined. The observation of right ascension gives us the meridian on which the body is situated in the heavens, and that of declination the precise point of that meridian, and the two combined give us the exact place of the body: for whenever we can ascertain the distance of a body in a known surface from a given point, measured in two directions perpendicular to each other (as in this case along the equator, and along a circle of declination), or, indeed, inclined at any given angle to each other, there is only one point which can fulfil both conditions, and that point therefore is completely ascertained.

In this manner we can determine the place of the sun on every day in the year; and the observation of all these points will give us so many of his successive positions in the heavens, and his path among the stars must, therefore, be a line passing through all these points. This line is found, when the observations are carefully made and registered, to be a great circle of the sphere, which intersects the equator at an angle of 23° 28' nearly, and is called the ecliptic*. All great circles of the sphere

The ecliptic being a great circle of the sphere, has, of course, two poles, which are distant from the poles of the equator by an arc of 23° 28′, equal to the inclination of the one circle to the other. If from the pole of the ecliptic a secondary be drawn through any heavenly body to the ecliptic, the arc of this circle between the body and the ecliptic is called the latitude of the body, and the arc of the ecliptic intercepted between the first point of Aries, and the intersection of the secondary with the ecliptic, is called the longitude of the body. The place of a body is of course as

C