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cardinal points, the axis of the earth, or to other principal astronomical lines and circles, which it ought to have to fulfil properly its objects.

(137.) Now, with respect to the first two classes of error, it must be observed, that, in so far as they cannot be reduced to known laws, and thereby become subjects of calculation and due allowance, they actually vitiate, to their full extent, the results of any observations in which they subsist. Being, however, in their nature casual and accidental, their effects necessarily lie sometimes one way, sometimes the other; sometimes diminishing, sometimes tending to increase the results. Hence, by greatly multiplying observations, under varied circumstances, by avoiding unfavourable, and taking advantage of favourable circumstances of weather, or otherwise using opportunity to advantage- and finally, by taking the mean or average of the results obtained, this class of errors may be so far subdued, by setting them to destroy one another, as no longer sensibly to vitiate any theoretical or practical conclusion. This is the great and indeed only resource against such errors, not merely to the astronomer, but to the investigator of numerical results in every department of physical research.

(138.) With regard to errors of adjustment and workmanship, not only the possibility, but the certainty of their existence, in every imaginable form, in all instruments, must be contemplated. Human hands or machines never formed a circle, drew a straight line, or erected a perpendicular, nor ever placed an instrument in perfect adjustment, unless accidentally; and then only during an instant of time. This does not prevent, however, that a great approximation to all these desiderata should be attained. But it is the peculiarity of astronomical observation to be the ultimate means of detection of all mechanical defects which elude by their minuteness every other mode of detection. What the eye cannot discern nor the touch perceive, a course of astronomical observations will make distinctly evident. The imperfect products of man's hands are here tested by being brought into comparison under very great magnifying powers (corresponding in effect to a great increase in acuteness of perception) with the perfect workmanship of nature; and there is none which will bear the trial. Now, it may seem like arguing in a vicious circle, to deduce theoretical conclusions and laws from observation, and then to turn round upon the instruments with which those observations were made, accuse them of imperfection, and attempt to detect and rectify their errors by means of the very laws and theories which they have helped us to a knowledge of. A little consideration, however, will suffice to show that such a course of proceeding is perfectly legitimate.

(139.) The steps by which we arrive at the laws of natural phenomena,

and especially those which depend for their verification on numerical determinations, are necessarily successive. Gross results and palpable laws are arrived at by rude observation with coarse instruments, or without any instruments at all, and are expressed in language which is not to be considered as absolute, but is to be interpreted with a degree of latitude commensurate to the imperfection of the observations themselves. These results are corrected and refined by nicer scrutiny, and with more delicate means. The first rude expressions of the laws which embody them are perceived to be inexact. The language used in their expression is corrected, its terms more rigidly defined, or fresh terms introduced, until the new state of language and terminology is brought to fit the improved state of knowledge of facts. In the progress of this scrutiny subordinate laws are brought into view which still further modify both the verbal statement and numerical results of those which first offered themselves to our notice; and when these are traced out and reduced to certainty, others, again, subordinate to them, make their appearance, and become subjects of further inquiry. Now, it invariably happens (and the reason is evident) that the first glimpse we catch of such subordinate laws the first form in which they are dimly shadowed out to our minds -is that of errors. We perceive a discordance between what we expect, and what we find. The first occurrence of such a discordance we attribute to accident. It happens again and again; and we begin to suspect our instruments. We then inquire, to what amount of error their determinations can, by possibility, be liable. If their limit of possible error exceed the observed deviation, we at once condemn the instrument, and set about improving its construction or adjustments. Still the same deviations occur, and, so far from being palliated, are more marked and better defined than before. We are now sure that we are on the traces of a law of nature, and we pursue it till we have reduced it to a definite statement, and verified it by repeated observation, under every variety of circumstances.

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(140.) Now, in the course of this inquiry, it will not fail to happen that other discordances will strike us. Taught by experience, we suspect the existence of some natural law, before unknown; we tabulate (i. e. draw out in order) the results of our observations; and we perceive, in this synoptic statement of them, distinct indications of a regular progression. Again we improve or vary our instruments, and we now lose sight of this supposed new law of nature altogether, or find it replaced by some other, of a totally different character. Thus we are led to suspect an instrumental cause for what we have noticed. We examine, therefore, the theory of our instrument; we suppose defects in its structure, and, by

the aid of geometry, we trace their influence in introducing actual errors into its indications. These errors have their laws, which, so long as we have no knowledge of causes to guide us, may be confounded with laws of nature, as they are mixed up with them in their effects. They are not fortuitous, like errors of observation, but, as they arise from sources inherent in the instrument, and unchangeable while it and its adjustments remain unchanged, they are reducible to fixed and ascertainable forms; each particular defect, whether of structure or adjustment, producing its own appropriate form of error. When these are thoroughly investigated, we recognize among them one which coincides in its nature and progression with that of our observed discordances. The mystery is at once solved. We have detected, by direct observation, an instrumental defect.

(141.) It is, therefore, a chief requisite for the practical astronomer to make himself completely familiar with the theory of his instruments. By this alone is he enabled at once to decide what effect on his observations any given imperfection of structure or adjustment will produce in any given circumstances under which an observation can be made. This alone also can place him in a condition to derive available and practical means of destroying and eliminating altogether 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

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 per formed, are, however, noticed, for the convenience of readers who may use sucn in struments without going farther into the arcana of practical astronomy.

difficulty. In fact, it is a case precisely analogous to 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 corrected-between 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 time-keepers, 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 23 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 23 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, independent 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 be

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