the influence of such imperfections, by so arranging his observations, that it shall affect their results in opposite ways, and that its influence shall thus disappear from their mean, which is one of the chief modes by which precision is attained in practical astronomy. Suppose, for example, the principle of an instrument required that a circle should be concentric with the axis on which it is made to turn. As this is a condition which no workmanship can exactly fulfil, it becomes necessary to inquire what errors will be produced in observations made and registered on the faith of such an instrument, by any assigned deviation in this respect; that is to say, what would be the disagreement between observations made with it and with one absolutely perfect, could such be obtained. Now, simple geometrical considerations suffice to show-1st. that if the axis be excentric by a given fraction (say one thousandth part) of the radius of the circle, all angles read off on that part of the circle towards which the excentricity lies, will appear by that fractional amount too small, and all on the opposite side too large. And, 2dly, that whatever be the amount of the excentricity, and on whatever part of the circle any proposed angle is measured, the effect of the error in question on the result of observations depending on the graduation of its circumference (or limb, as it is technically called) will be completely annihilated by the very easy method of always reading off the divisions on two diametrically opposite points of the circle, and taking a mean; for the effect of excentricity is always to increase the arc representing the angle in question on one side of the circle, by just the same quantity by which it diminishes that on the other. Again, suppose that the proper use of the instrument required that this axis should be exactly parallel to that of the earth. As it never can be placed or remain so, it becomes a question, what amount of error will arise, in its use, from any assigned deviation, whether in a horizontal or vertical plane, from this precise position. Such inquiries constitute the theory of instrumental errors; a theory of the utmost importance to practice, and one of which a complete knowledge will enable an observer, with moderate instrumental means, often to

attain a degree of precision which might seem to belong only to the most refined and costly. This theory, as will readily be apprehended, turns almost entirely on considerations of pure geometry, and those for the most part not difficult. In the present work, however, we have no further concern with it. The astronomical instruments we propose briefly to describe in this chapter will be considered as perfect both in construction and adjustment.*

(142.) As the above remarks are very essential to a right understanding of the philosophy of our subject and the spirit of astronomical methods, we shall elucidate them by taking one or two special cases. Observant persons, before the invention of astronomical instruments, had already concluded the apparent diurnal motions of the stars to be performed in circles about fixed poles in the heavens, as shown in the foregoing chapter. In drawing this conclusion, however, refraction was entirely overlooked, or, if forced on their notice by its great magnitude in the immediate neighbourhood of the horizon, was regarded as a local irregularity, and, as such, neglected, or slurred over. As soon, however, as the diurnal paths of the stars were attempted to be traced by instruments, even of the coarsest kind, it became evident that the notion of exact circles described about one and the same pole would not represent the phenomena correctly, but that, owing to some cause or other, the apparent diurnal orbit of every star is distorted from a circular into an oval form, its lower segment being flatter than its upper; and the deviation being greater the nearer the star approached the horizon, the effect being the same as if the circle had been squeezed upwards from below, and the lower parts more than the higher. For such an effect, as it was soon found to arise from no casual or instrumental cause, it became necessary to seek a natural one; and refraction readily occurred, to solve the difficulty. In fact, it is a case precisely analogous to

The principle on which the chief adjustments of two or three of the most useful and common instruments, such as the transit, the equatorial, and the sextant, are performed, are, however, noticed, for the convenience of readers who may use such instruments without going farther into the arcana of practical astronomy.

what we have already noticed (art. 47.), of the apparent distortion of the sun near the horizon, only on a larger scale, and traced up to greater altitudes. This new law once established, it became necessary to modify the expression of that anciently received, by inserting in it a salvo for the effect of refraction, or by making a distinction between the apparent diurnal orbits, as affected by refraction, and the true ones cleared of that effect. This distinction between the apparent and the true-between the uncorrected and correctedbetween the rough and obvious, and the refined and ultimate is of perpetual occurrence in every part of astronomy.

(143.) Again. The first impression produced by a view of the diurnal movement of the heavens is that all the heavenly bodies perform this revolution in one common period, viz. a day, or 24 hours. But no sooner do we come to examine the matter instrumentally, i. e. by noting, by timekeepers, their successive arrivals on the meridian, than we find differences which cannot be accounted for by any error of observation. All the stars, it is true, occupy the same interval of time between their successive appulses to the meridian, or to any vertical circle; but this is a very different one from that occupied by the sun. It is palpably shorter; being, in fact, only 23h 56' 4.09", instead of 24 hours, such hours as our common clocks mark. Here, then, we have already two different days, a sidereal and a solar; and if, instead of the sun, we observe the moon, we find a third, much longer than either, a lunar day, whose average duration is 24h 54m of our ordinary time, which last is solar time, being of necessity conformable to the sun's successive re-appearances, on which all the business of life depends.

(144.) Now, all the stars are found to be unanimous in giving the same exact duration of 23h 56' 4.09", for the sidereal day; which, therefore, we cannot hesitate to receive as the period in which the earth makes one revolution on its axis. We are, therefore, compelled to look on the sun and moon as exceptions to the general law; as having a different nature, or at least a different relation to us, from the stars; and as having motions, real or apparent, of their own, inde

pendent of the rotation of the earth on its axis. Thus a great and most important distinction is disclosed to us.

(145.) To establish these facts, almost no apparatus is required. An observer need only station himself to the north of some well-defined vertical object, as the angle of a building, and, placing his eye exactly at a certain fixed point (such as a small hole in a plate of metal nailed to some immoveable support), notice the successive disappearances of any star behind the building, by a watch. When he observes the sun, he must shade his eye with a dark-coloured or smoked glass, and notice the moments when its western and eastern edges successively come up to the wall, from which, by taking half the interval, he will ascertain (what he cannot directly observe) the moment of disappearance of its

centre.

(146.) When, in pursuing and establishing this general fact, we are led to attend more nicely to the times of the daily arrival of the sun on the meridian, irregularities (such they first seem to be) begin to make their appearance. The intervals between two successive arrivals are not the same at all times of the year. They are sometimes greater, sometimes less, than 24 hours, as shown by the clock; that is to say, the solar day is not always of the same length. About the 21st of December, for example, it is half a minute longer, and about the same day of September nearly as much shorter, than its average duration. And thus a distinction is again pressed upon our notice betwen the actual solar day, which is never two days in succession alike, and the mean solar day of 24 hours, which is an average of all the solar days throughout the year. Here, then, a new source of inquiry opens to us. The sun's apparent motion is not only not the

* This is an excellent practical method of ascertaining the rate of a clock or watch, being exceedingly accurate if a few precautions are attended to; the chief of which is, to take care that that part of the edge behind which the star (a bright one, not a planet) disappears shall be quite smooth; as otherwise variable refraction may transfer the point of disappearance from a protuberance to a notch, and thus vary the moment of observation unduly. This is easily secured, by nailing up a smooth-edged board. The verticality of its edge should be ensured by the use of a plumb-line.

same with that of the stars, but it is not (as the latter is) uniform. It is subject to fluctuations, whose laws become matter of investigation. But to pursue these laws, we require nicer means of observation than what we have described, and are obliged to call in to our aid an instrument called the transit instrument, especially destined for such observations, and to attend minutely to all the causes of irregularity in the going of clocks and watches which may affect our reckoning of time. Thus we become involved by degrees in more and more delicate instrumental inquiries; and we speedily find that, in proportion as we ascertain the amount and law of one great or leading fluctuation, or inequality, as it is called, of the sun's diurnal motion, we bring into view others continually smaller and smaller, which were before obscured, or mixed up with errors of observation and instrumental imperfections. In short, we may not inaptly compare the mean length of the solar day to the mean or average height of water in a harbour, or the general level of the sea unagitated by tide or waves. The great annual fluctuation above noticed may be compared to the daily variations of level produced by the tides, which are nothing but enormous waves extending over the whole ocean, while the smaller subordinate inequalities may be assimilated to waves ordinarily so called, on which, when large, we perceive lesser undulations to ride, and on these, again, minuter ripplings, to the series of whose subordination we can perceive no end. - (147.) With the causes of these irregularities in the solar motion we have no concern at present; their explanation belongs to a more advanced part of our subject: but the distinction between the solar and sidereal days, as it pervades every part of astronomy, requires to be early introduced, and never lost sight of. It is, as already observed, the mean or average length of the solar day, which is used in the civil reckoning of time. It commences at midnight, but astronomers, even when they use mean solar time, depart from the civil reckoning, commencing their day at noon, and reckoning the hours from 0 round to 24. Thus, 11 o'clock in the forenoon of the second of January, in the civil reckon