cipal classes. The first comprehends those which arise from an instrument not being what it professes to be, which is error of workmanship. Thus, if a pivot or axis, instead of being, as it ought, exactly cylindrical, be slightly flattened, or elliptical, if it be not exactly (as it is intended it should) concentric with the circle it carries ;—if this circle (so called) be in reality not exactly circular, or not in one plane ;-if its divisions, intended to be precisely equidistant, should be placed in reality at unequal intervals,—and a hundred other things of the same sort. These are not mere speculative sources of error, but practical annoyances, which every observer has to contend with.

(108.) The other subdivision of instrumental errors comprehends such as arise from an instrument not being placed in the position it ought to have; and from those of its parts, which are made purposely moveable, not being properly disposed inter se. These are errors

of adjustment. Some are unavoidable, as they arise from a general unsteadiness of the soil or building in which the instruments are placed; which, though too minute to be noticed in any other way, become appreciable in delicate astronomical observations: others, again, are consequences of imperfect workmanship, as where an instrument once well adjusted will not remain so, but keeps deviating and shifting. But the most important of this class of errors arise from the nonexistence of natural indications, other than those afforded by astronomical observations themselves, whether an instrument has or has not the exact position, with respect to the horizon and its 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.

(109.) 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 observa

CHAP. II. MUTUAL DESTRUCTION OF ERRORS.

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tions 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, and taking the mean or average of their results, 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.

(110.) 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 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 observ. ation, 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 consi

deration, however, will suffice to show that such a course of proceeding is perfectly legitimate.

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(111.) 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 these are corrected and refined upon by nicer scrutiny with more delicate means. In the progress of this, subordinate laws are brought into view, which 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 enquiry. 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 enquire, 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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(112.) Now, in the course of this enquiry, 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 per

CHAP. II. DETECTION OF INSTRUMENTAL ERRORS. 71

ceive, 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, and 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.

(113.) It is, therefore, a chief requisite for the practical astronomer to make himself completely familiar with the theory of his instruments, so as to be able 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. Suppose, for example, that the principle of an instrument required that a circle should be exactly concentric with the axis on which it is made to turn. As this is a condition which no workmanship can fulfil, it becomes necessary to enquire 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 dis

agreement between observations made with it and with one absolutely perfect, could such be obtained. Now, a simple theorem in geometry shows that, whatever be the extent of this deviation, it may be annihilated in its effect on the result of observations depending on the graduation of the limb, by the very easy method of 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 one such reading by just the same quantity by which it diminishes 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 enquiries 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 very moderate instrumental means, to attain a degree of precision which might seem to belong only to the most refined and costly. In the present work, however, we have no further concern with it. The few astronomical instruments we propose to describe in this chapter will be considered as perfect both in construction and adjustment.

(114.) 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 a case. 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