the difference of direction of two lines is the definition of the angle which they form with each other, parallax is therefore the angle subtended at the distant object by the radius of the earth. Observations made at different parts of the earth's surface, when corrected for parallax, all become as if made from the centre of the earth, and thus admit of direct comparison with one another, for the purpose of indicating the precise angular movement of the object itself.

Passing over the second chapter we come to the third, which is one of the best in the book. It explains the nature and use of the astronomical instruments. In the introduction to this chapter is an admirable exposition, in a general way, of the means by which the astronomer, with the necessarily imperfect workmanship of the most skilful human hands, arrives at perfect results in his observations.

"Astronomical instrument-making may be justly regarded as the most refined of the mechanical arts, and that in which the nearest approach to geometrical precision is required, and has been attained. It may be thought an easy thing, by one unacquainted with the niceties required, to turn a circle in metal, to divide its circumference into three hundred and sixty equal parts, and these again into smaller subdivisions,-to place it accurately on its centre, and to adjust it in a given position; but practically it is found to be one of the most difficult. Nor will this appear extraordinary, when it is considered that, owing to the application of telescopes to the purposes of angular measurement, every imperfection of structure of division becomes magnified by the whole optical power of that instrument; and that thus, not only direct errors of workmanship, arising from unsteadiness of hand or imperfection of tools, but those inaccuracies which originate in far more uncontrollable causes, such as the unequal expansion and contraction of metallic masses, by a change of temperature, and their unavoidable flexure or bending by their own weight, become perceptible and measurable. An angle of one minute occupies, on the circumference of a circle of ten inches in radius, only about th part of an inch, a quantity too small to be certainly dealt with without the use of magnifying glasses; yet one minute is a gross quantity in the astronomical measurement of an angle. With the instruments now employed in observatories, a single second, or the 60th part of a minute, is rendered a distinctly visible and appreciable quantity. Now, the arc of a circle, subtended by one second, is less than the 200,000th part of the radius, so that on a circle of six feet in diameter it would occupy no greater linear extent thanth part of an inch; a quantity requiring a powerful microscope to be discerned at all. ever-varying fluctuations of heat and cold have a tendency to produce not merely temporary and transient, but permanent, uncompensated changes of form in all considerable masses of those metals which alone are applicable to such uses; and their own weight, however symmetrically formed, must always be unequally sustained, since it is impossible to apply the sustaining power to every part separately: even could this be done, at all events force must be used to move and to fix them; which can never be done without producing temporary, and risking permanent, change of form. It is true, by dividing them on their centres, and in the identical places they are destined to occupy, and by a thousand ingenious and delicate contrivances, wonders have been accomplished in this department of art, and a degree of perfection has been given, not merely to chefs d'œuvre,

The

but to instruments of moderate prices and dimensions, and in ordinary use, which, on due consideration, must appear very surprising. But though we are entitled to look for wonders at the hands of scientific artists, we are not to expect miracles. The demands of the astronomer will always surpass the power of the artist; and it must, therefore, be constantly the aim of the former to make himself, as far as possible, independent of the imperfections incident to every work the latter can place in his hands. He must, therefore, endeavour so to combine his observations, so to choose his opportunities, and so to familiarize himself with all the causes which may produce instrumental derangement, and with all the peculiarities of structure and material of each instrument he possesses, as not to allow himself to be misled by their errors, but to extract from their indications, as far as possible, all that is true, and reject all that is erroneous. It is in this that the art of the practical astronomer consists, an art of itself of a curious and intricate nature, and of which we can here only notice some of the leading and general features.

"The great aim of the practical astronomer being numerical correctness in the results of instrumental measurement, his constant care and vigilance must be directed to the detection and compensation of errors, either by annihilating, or by taking account of, and allowing for them. Now, if we examine the sources from which errors may arise in any instrumental determination, we shall find them chiefly reducible to three principal heads:

“1st. External or incidental causes of error; comprehending such as depend on external, uncontrollable circumstances: such as fluctuations of weather, which disturb the amount of refraction from its tabulated value, and being reducible to no fixed law, induce uncertainty to the extent of their own possible magnitude; such as, by varying the temperature of the air, vary also the form and position of the instruments used, by altering the relative magnitudes and the tension of their parts; and others of the like nature.

"2dly. Errors of observation: such as arise, for example, from inexpertness, defective vision, slowness in seizing the exact instant of occurrence of a phenomenon, or precipitancy in anticipating it, &c.; from atmospheric indistinctness; insufficient optical power in the instrument, and the like. Under this head may also be classed all errors arising from momentary instrumental derangement, slips in clamping, looseness of screws, &c.

"3dly. The third, and by far the most numerous class of errors to which astronomical measurements are liable, arise from causes which may be deemed instrumental, and which may be subdivided into two principal 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.

"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 movable, 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 non-existence of natural indications, other than those afforded by astronomical observations themselves. 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.

"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 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."-P. 85-89, §§ 131-138.

Here follows the passage which we have already quoted: (page 32, ante.)

Certain classes of instrumental error may be known and provided for a priori. Such as arise from the foreseen impossibility of perfection in some particular of the construction.

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 eccentric 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 eccentricity 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 eccentricity, 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 eccentricity 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.”—P. 91, § 141.

An easily apprehended elucidation of the spirit of astronomic method is given in connexion with the subject we are now upon:

"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. 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."-P. 92, § 142.

Another illustration of the incorrectness of first impressions is to be found in the time of diurnal revolutions of the sun, moon, and stars:

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, namely, a day, or twenty-four hours. But no sooner do we come to examine the matter instrumentally, that is, 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 23h 56′ 4.09′′, instead of twenty-four 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.

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 differ

ent 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.

"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 immovable support,) notice the successive disappearances of any star behind the building, by a watch."-P. 93, § 143, 145.

Exact observations upon the length of the sidereal day are best made by noting the instant at which a star culminates, or makes its transit over the meridian or great circle which passes through the poles and zenith of a place. The observation is taken with what is called a transit instrument, which is a telescope made with projections from the sides of the tube resembling the trunnions of a cannon, and called the supporting axis of the instrument; and, as a cannon is supported by the trunnions resting on the cheeks of the carriage, so the telescope is by the extremities of its supporting axis resting upon two stone piers, one of which is on the east, the other on the west side of the telescope, which points north and south; and, as it turns upon its trunnions, the line of vision, as one looks through the telescope, prolonged to the celestial sphere or blue vault of the sky, describes upon the latter the circle which is the celestial meridian of the place.

This instrument is used in connexion with what is called the astronomical clock, keeping sidereal time. A common clock, with the pendulum a little shortened, will answer the purpose; but the clock, to be as useful as possible, should be of the best workmanship, and have a mercurial or other compensating pendulum, which does not change its length with changes of temperature.

The zero of time by the astronomical clock, is the instant that the point of the heavens known as the vernal equinoctial point is on the meridian. This is the point in which the annual path of the sun amongst the stars crosses the celestial equator. When any star is upon the meridian, the astronomical clock will show the right ascension of the star in time; that is, the difference in time between the meridian transits of the vernal equinox and the meridian transit of the star. The right ascension in time being multiplied by 15, gives the right ascension in space, or in degrees and fractions of a degree. Because, as the earth turns round in 24 hours, 360° ÷ 24°, or 15°, will be the space that the earth turns in one hour.

The transit instrument is used by elevating the telescope till the altitude shown on a graduated circle at one extremity of the supporting axis is equal to the meridian altitude of the star, which depends