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which is the true horizon, and though mathematically speaking no two plumb-lines are exactly parallel (since they converge to the earth’s centre), yet over very small tracts, such as the area of a building — in one and the same town, &c., the difference from exact parallelism is so small that it may be practically disregarded.' To a spectator looking upwards such a system of plumb-lines will appear to converge to his zenith ; downwards, to his nadir.

(117.) So also the celestial equator, or the equinoctial, must be conceived as the vanishing circle of a system of planes parallel to the earth's equator, or perpendicular to its axis. The celestial horizon of any spectator is in like manner the vanishing circle of all planes parallel to his true horizon, of which planes his rational horizon (passing through the earth's centre) is one, and his sensible horizon (the tangent plane of his station) another.

(118.) Owing, however, to the absence of all the ordinary indications of distance which influence our judgment in respect of terrestrial objects, owing to the want of determinate figure and magnitude in the stars and planets as commonly seen — the projection of the celestial bodies on the ground of the heavenly concave is not usually regarded in this its true light, of a perspective representation or picture, and it even requires an effort of imagination to conceive them in their true relations, as at vastly different distances, one behind the other, and forming with one another lines of junction violently foreshortened, and including angles altogether differing from those which their projected representations appear to make. To do so at all with effect presupposes a knowledge of their actual situations in space, which it is the business of astronomy to arrive at by appropriate considerations. But the connections which subsist among the several parts of the picture, the purely geometrical relations among the angles and sides of the spherical triangles of which it consists, constitute, under the name of Uranometry,' a preliminary and subordinate branch of the general science, with which it is necessary to be familiar before any further progress can be made. Some of the most elementary and frequently occurring of these relations we proceed to explain. And first, as immediate consequences of the above definitions, the following propositions will be borne in mind.

(119.) The altitude of the elevated pole is equal to the latitude of the spectator's geographical station.

For it appears, see fig. art. 112, that the angle PA Z between the 1 An interval of a mile corresponds to a convergence of plumb-lines amounting to somewhal sess space than a minute.

Oupavos, the heavens; perpev, to measure: the measurement of the heavens.

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pole and the zenith is equal to NCA, and the angles Z A n and NCE being right angles, we have P A n=A C E. Now the former of these is the elevation of the pole as seen from E, the latter is the angle at the earth's centre subtended by the arc E A, or the latitude of the place.

(120.) Hence to a spectator at the north pole of the earth, the north pole of the heavens is in his zenith. As he travels southward it becomes less and less elevated till he reaches the equator, when both poles are in his horizon — south of the equator the north pole becomes depressed below, while the south rises above his horizon, and continues to do so till the south pole of the globe is reached, when that of the heavens will be in the zenith.

(121.) The same stars, in their diurnal revolution, come to the meridian, successively, of every place on the globe once in twenty-four sidereal hours. And, since the diurnal rotation is uniform, the interval, in sidereal time, which elapses between the same star coming upon the meridians of two different places is measured by the difference of longitudes of the places.

(122.) Vice versa the interval elapsing between two different stars coming on the meridian of one and the same place, expressed in sidereal time, is the measure of the difference of right ascensions of the stars.

(123.) The equinoctial iptersects the horizon in the east and west points, and the meridian a à point whose altitude is equal to the co-latitude of the place. Thus, at Greenwich, of which the latitude is 51° 28' 40", the altitude of the intersection of the equinoctial and meridian is 38° 31' 20". The north and south poles of the heavens are the poles of the equinoctial. The east and west points of the horizon of a spectator are the poles of his celestial meridian. The north and south points of his horizon are the poles of his prime vertical, and his zenith and nadir are the poles of his horizon.

(124.) All the heavenly bodies culminate (i. e. come to their greatest altitudes) on the meridian ; which is, therefore, the best situation to observe them, being least confused by the inequalities and vapours of the atmosphere, as well as least displaced by refraction.

(125.) All celestial objects within the circle of perpetual apparition come twice on the meridian, above the horizon, in every diurnal revolution; once above and once below the pole. These are called their upper and lourer culminations.

(126.) The problems of uranometry, as we have described it, consist in the solution of a variety of spherical triangles, both right and oblique angled, according to the rules, and by the formula of spherical trigononetry, which we suppose known to the reader, or for which he will consult appropriate treatises. We shall only here observe generally, that in all

problems in which spherical geometry is concerned, the student will find it a useful practical maxim rather to consider the poles of the great circles which the question before him refers to than the circles themselves. То use, for example, in the relations he has to consider, polar distances rather than declinations, zenith distances rather than altitudes, &c. Bearing 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 ZPS, Z is the zenith, P the elevated pole, and S the star, sun, or other celestial object. In this triangle occur, 1st, PZ, which being the complement of PH (the altitude of the pole), is obviously the complement of the latitude (or the co-latitude, as it is called) of the place; 2d, PS, the polar distance, or the complement of the declination (co-declination) of the star; 30, ZS, the zenith distance or co-altitude of the star. If PS be greater than 90°, the object is situated on the side of the equinoctial opposite to that of the elevated pole. If ZS be so, the object is below the horizon.

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In the same triangle the angles are, 1st, Z PS the lower angle; 2d, PZS (the supplement of SZ0, which latter is the azimuth of the star or other heavenly body), 3d, PS Z, an angle which, from the infrequency of any practical reference to it, has not acquired a name.'

- 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 following five astronomical magnitudes, then, occur among the sides 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 31' 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 Z P of the place, we have given 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 Z P S, the polar distance P S of the star, and Z S, its coaltitude 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 observation; 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 OBSERVATIONS

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 INSTRUMENTS. 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 principal means which astronomers

· 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.

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