ing this in mind, there are few problems in uranometry which will offer any difficulty. The following are the combinations which most commonly occur for solution when the place of one celestial object only on the sphere is concerned. (127.) In the triangle Z PS, Z is the zenith, P the elevated pole, and S the star, sun, or other celestial object. In this triangle occur, 1st, P Z, which being the complement of P H (the altitude of the pole), is obviously the complement of the latitude (or the co-latitude, as it is called) of the place; 2d, P S, the polar distance, or the complement of the declination (co-declination) of the star; 3d, Z S, the zenith distance or co-altitude of the star. If P S be greater than 90°, the object is situated on the side of the equinoctial opposite to that of the elevated pole. If Z S be so, the ob In the ow the horizon elevated pole. In the same triangle the angles are, 1st, Z P S the lower angle; 2d, PZS (the supplement of SZO, which latter is the azimuth of the star or other heavenly body), 3d, P SZ, an angle which, from the infrequency of any practical reference to it, has not acquired a name.* The following five astronomical magnitudes, then, occur among the sides and angles of this most useful triangle: viz. 1st, The co-latitude of the place of observation ; 2d, the polar distance ; 3d, the zenith distance; 4th, the hour angle; and 5th, the sub-azimuth (supplement of azimuth) of a given celestial object; and by its solution therefore may all problems be resolved, in which three of these magnitudes are directly or indirectly given, and the other two required to be found. (128.) For example, suppose the time of rising or setting of the sun or of a star were required, having given its right ascension and polar distance. The star rises when apparently on the horizon, or really about 34' below it (owing to refraction), so that, at the moment of its apparent rising, its zenith distance is 90° 34'=Z S. Its polar distance PS being also given, and the co-latitude ZP of the place, we have given * In the practical discussion of the measures of double stars and other objects by the aid of the position micrometer, this angle is sometimes required to be known; and, when so required, it will be not inconveniently referred to as the angle of position of the zenith." the three sides of the triangle, to find the hour angle ZPS, which, being known, is to be added to or subtracted from the star's right ascension, to give the sidereal time of setting or rising, which, if we please, may be converted into solar time by the proper rules and tables. (129.) As another example of the use of the same triangle, we may propose to find the local sidereal time, and the latitude of the place of observation, by observing equal altitudes of the same star east and west of the meridian, and noting the interval of the observations in sidereal time. The hour angles corresponding to equal altitudes of a fixed star being equal, the hour angle east or west will be measured by half the observed interval of the observations. In our triangle, then, we have given this hour angle ZPS, the polar distance P S of the star, and Z S, its co-altitude at the moment of observation. Hence we may find P Z, the co-latitude of the place. Moreover, the hour angle of the star being known, and also its right ascension, the point of the equinoctial is known, which is on the meridian at the moment of observaation; and, therefore, the local sidereal time at that moment. This is a very useful observation for determining the latitude and time at an unknown station. CHAPTER III.* OF THE NATURE OF ASTRONOMICAL INSTRUMENTS AND OBSERVA TIONS IN GENERAL. — OF SIDEREAL AND SOLAR TIME. — OF THE MEASUREMENTS OF TIME. - CLOCKS, CHRONOMETERS. — OF ASTRONOMICAL MEASUREMENTS. - PRINCIPLE OF TELESCOPIC SIGHTS TO INCREASE THE ACCURACY OF POINTING. - SIMPLEST APPLICATION OF THIS PRINCIPLE. THE TRANSIT INSTRUMENT. -OF THE MEASUREMENT OF ANGULAR INTERVALS. — METHODS OF INCREASING THE ACCURACY OF READING. — THE VERNIER. — THE MICROSCOPE. — OF THE MURAL CIRCLE. -- THE MERIDIAN CIRCLE. — FIXATION OF POLAR AND HORIZONTAL POINTS. — THE LEVEL, PLUMB-LINE, ARTIFICIAL HORIZON. — PRINCIPLE OF COLLIMATION. - COLLIMATORS OF RITTENHOUSE, KATER, AND BENZENBERG. — OF COMPOUND INSTRUMENTS WITH CO-ORDINATE CIRCLES. THE EQUATORIAL, ALTITUDE, AND AZIMUTH INSTRUMENT. — THEODOLITE. — OF THE SEXTANT AND REFLECTING CIRCLE. — PRINCIPLE OF REPETITION. — OF MICROMETERS. - PARALLEL WIRE MICROMETER. — PRINCIPLE OF THE DUPLICATION OF IMAGES. — THE HELIOMETER. DOUBLE REFRACTING EYE-PIECE. — VARIABLE PRISM MICROMETER. OF THE POSITION MICROMETER. (130.) OUR first chapters have been devoted to the acquisition chiefly of preliminary notions respecting the globe we inhabit, its relation to the celestial objects which surround it, and the physical circumstances under which all astronomical observations must be made, as well as to provide ourselves with a stock of technical words and elementary ideas of most frequent and familiar use in the sequel. We might now proceed to a more exact and detailed statement of the facts and theories of astronomy; but, in order to do this with full effect, it will be desirable that the reader be made acquainted with the • The student who is anxious to become acquainted with the chief subject matter of this work, may defer the reading of that part of this chapter which is devoted to the description of particular instruments, or content himself with a cursory perusal of it, until farther advanced, when it will be necessary to return to it. principal means which astronomers possess, of determining, with the degree of nicety their theories require, the data on which they ground their conclusions; in other words, of ascertaining by measurement the apparent and real magnitudes with which they are conversant. It is only when in possession of this knowledge that he can fully appretiate either the truth of the theories themselves, or the degree of reliance to be placed on any of their conclusions antecedent to trial : since it is only by knowing what amount of error can certainly be perceived and distinctly measured, that he can satisfy himself whether any theory offers so close an approximation, in its numerical results, to actual phenomena, as will justify him in receiving it as a true representation of nature. (131.) 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 in to 360 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 10 inches in radius, only about z oth 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 appretiable 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 6 feet in diameter it would occupy no greater linear extent than both part of an inch; a quantity requiring a powerful microscope to be discerned at all. Let any one figure to himself, therefore, the difficulty of placing on the circumference of a metallic circle of such dimensions (supposing the difficulty of its construction surmounted), 360 marks, dots, or cognizable divisions, which shall all be true to their places within such narrow limits; to say nothing of the subdivision of the degrees so marked off into minutes, and of these again into seconds. Such a work has probably baffled, and will probably for ever continue to baffle, the utmost stretch of human skill and industry; nor, if executed, could it endure. The 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'æuvre, 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 |