398. Methods of computing the sides of terrestrial triangles 399. Formulæ for computing a side of a triangle in feet or in - 400. Formula for the reduction of a spherical arc to its chord 401. Rule for the reduction of a spherical angle to the angle 402. Theorem of Legendre for computing the sides of terres- 403. Manner of determining the position of a meridian line by 405. Method of determining the azimuth of a terrestrial object 407. Method of reducing computed portions of the meridian to 409. Method of finding the latitude of a station by observed 410. Investigations of formulæ for determining the differences between the latitudes and longitudes of two stations - 378 411. Proof, from the lengths of the degrees of latitude, that the NOTE.—The following propositions relate to the values of terrestrial arcs on the supposition that the earth is a spheroid of revolution. 412. (Prop. I.) Every section of a spheroid of revolution when made by a plane oblique to the equator is an ellipse 413. (Prop. II.) The excess of the angles of a terrestrial triangle 414. (Prop. III.) To investigate the radius of curvature for a vertical section of a spheroid 415. (Prop. IV.) To investigate the ratio between the earth's 416. (Prop. V.) To investigate the law of the increase of the degrees of latitude from the equator towards the poles - 388 417. (Prop. VI.) To determine the radius, and length of an arc on any parallel of terrestrial latitude 418. (Prop. VII.) To find the ratio between the earth's axes, 419. (Prop. VIII.) To find the distance in feet, on an elliptical meridian, between a vertical arc perpendicular to the me- ridian and a parallel of latitude, both passing through a given point. Also, to find the difference in feet between the vertical arc and the corresponding portion of the 420. (Prop. IX.) To investigate an expression for the length of a meridional arc on the terrestrial spheroid; having, by ob- servation, the latitudes of the extreme points, with as- 421. Table of the measured lengths of a degree of latitude in different places. Presumed value of the earth's ellipticity 391 422. The triangles in a geodetical survey gradually increase as they 423. Methods of determining the stations in the secondary triangles 393 424. (Prob. I.) To determine the positions of two objects and the distance between them, when there have been observed, at those objects, the angles contained between the line joining them and lines imagined to be drawn from them to two stations whose distance from each other is known - 394 425. (Prob. II.) To determine the position of an object, when there have been observed the angles contained between lines imagined to be drawn from it to three stations whose mutual distances are known. Use of the station-pointer - 395 426. (Prob. III.) To determine the positions of two objects with respect to three stations whose mutual distances are known, by angles observed as in the former propositions; and some one of the stations being invisible from each object 398 427. Investigation of a formula for correcting the computed heights of mountains, on account of the earth's curvature 399 428. The height of the mercury in a barometer applicable to the 431. Nature of mountain barometers. Formula for finding the relative heights of stations by the thermometrical barometer 404 432. The vibrations of pendulums in different regions serve to - 434. Manner of making experiments with detached pendulums · 407 NOTE. The following propositions relate to the corrections which are 435. (Prop. I.) To reduce the number of vibrations made on a 436. (Prop. II.) To correct the length of a pendulum on account · - - - - 437. (Prop. III.) To correct the length of a pendulum on account 438. (Prop. IV.) To correct the length of a pendulum on account - - 439. Formulæ for determining the length of a seconds pendulum at the equator, the ellipticity of the earth, and the varia- 440. The elements of terrestrial magnetism 441. The usual variation compass, and dipping needle 442. Mayer's dipping needle - Page 443. Formulæ for finding the dip, or inclination, of a needle when 444. Correspondence of the theory of magnetized needles to that observed declination on account of torsion 447. Formulæ for determining from observation the horizontal intensity of terrestrial magnetism 448. The horizontal force and vertical force magnetometer. For- mulæ for determining the ratios which the variations of the horizontal and vertical intensities bear to those compo- 449. Manner of expressing the intensity of terrestrial magnetism 423 450. Equations of condition exemplified, for obtaining the most probable values of elements. For the angles which, with their sum or difference, have been observed. For the transits of stars. For the length of a meridional arc. PRACTICAL ASTRONOMY AND GEODESY. CHAPTER I. THE EARTH.-PHENOMENA OF THE CELESTIAL BODIES. FORM OF THE EARTH, AND ROTATION ON ITS AXIS. APPARENT MOVEMENT OF THE STARS. REVOLUTION OF THE MOON ABOUT THE EARTH. HYPOTHESIS OF THE EARTH'S ANNUAL MOTION. PHASES OF THE MOON. -APPARENT MOVEMENTS OF THE PLANETS. THE CIRCLES OF THE SPHERE. 1. THAT the surface of the earth is of a form nearly spherical may be readily inferred from the appearance presented at any point on the ocean by a ship when receding from thence; for, on observing that the line which bounds the view on all sides is accurately or nearly the circumference of a circle, and that when a ship has reached any part of this line she seems to sink into the water, the spectator recognizes the fact that she is moving on a surface to which the visual rays from that circumference are tangents. These rays may be imagined to constitute the surface of a cone of which the eye of the spectator is the vertex; and the solid with which, at every part of its surface, a cone is in contact on the periphery of a line which is accurately or nearly a circle (that is, the solid whose section when cut any where by a plane is accurately or nearly a circle) is (Geom. 1. Prop. Cylind.) accurately or nearly a sphere. The like inference may be drawn from the appearance presented on all sides of a spectator on land, the curve line which bounds his view being the circumference of a circle except where inequalities of the ground destroy its regularity. 2. The plane of the circle which terminates the view of a spectator is designated his visible or sensible horizon. A plane B conceived to pass through the spectator and the sun at noon, perpendicularly to the horizon, is called his meridian; and, on the supposition that the earth is a sphere or spheroid, this plane will pass through its centre. Its intersection with the surface of the earth or with a horizontal plane, which, to the extent of a few yards in every direction about the spectator, may be considered as coincident with that surface, is called a meridian line: of this line, the extremity which is nearest to the Arctic regions of the earth is called the north point, and that which is opposite to it, the south point. A line imagined to pass through the spectator perpendicularly to the plane of the horizon, and to be produced above and below it towards the heavens, is denominated a vertical line; its upper and lower extremities are designated, respectively, the zenith and nadir. Every plane which may be conceived to pass through this line is said to be a vertical plane, but that which is at right angles to the plane of the meridian is called the prime vertical: it cuts the plane of the horizon in a line whose extremities are called the east and west points; the former being that which is on the right hand of the spectator when he looks towards the Arctic regions of the earth, and the latter, that which is on his left hand when in the same position. 3. Now, if a spectator were at any season of the year to land on the shores of Spitzbergen, the stars which are visible would appear to describe about him circles nearly parallel to his horizon. In the British Isles certain stars towards the north indicate by their movements that they describe during a day and a night the circumferences of circles whose planes are very oblique to the horizon and wholly above it, while others describe arcs which are easily seen to become smaller portions of a circumference as they rise more remotely from the northern part of the horizon; and a few may be observed which rise and set near the southern point, describing, during the time they are visible, curves which ascend but little above that plane. About the mouth of the Amazon, and in the islands of the Indian Ocean, the spectator would see the stars rise and set perpendicularly to the horizon, each of them describing half the circumference of a circle above it. If the spectator were to transfer himself to the southern regions of the earth he would see phenomena similar to those above mentioned exhibited by the stars which are situated in that part of the heavens; while on directing his eye towards the north, the stars which before were seen to ascend to considerable heights above the southern part of the horizon, would be either invisible or would be seen but for a short time, the places of rising and setting being near the northern point. |