PREFACE TO THE FOURTH EDITION. THE great and steadily increasing favour bestowed on this work, as modified by its author in 1842, would seem to indicate that no further changes were needed, or would be acceptable to teachers: the undersigned has therefore, in revising the third edition for the press, thought it best to confine himself principally to a correction of typographical, and other errors; and to such additions as the recent progress of the science demanded. Some additions have been made in the chapter on instruments; and, throughout the First Part, such changes and modifications of the formulæ and demonstrations introduced, as have been dictated by eight years' experience in the use of this work as a class book. The tables of the elements of the planetary orbits, have been arranged in a more convenient form, and extended so as to include those of the new planets, as far as they are at present known. These elements have been invariably derived from the most reliable sources. In the Second Part, very many inaccuracies have been corrected, and several problems and examples of a practical nature inserted. In connection with one of these problems, a table of reductions to the meridian has been given at the end of the book. With this exception, and that of an alteration in Table IX., adapting it to the present time, the tables are the same as those in the third edition. E. O. KENDALL. Central High School, Philadelphia, February 1, 1851. PREFACE TO THE THIRD EDITION. IN preparing this edition, the greater part has been written anew and so modified as to increase the value of the work as a text book. Several of the more abstruse investigations of the First Part have been omitted, references being made to larger works, and others have been transferred to the Appendix. Many of the figures illustrating the text have been rendered more perspicuous by a change in the construction, and a number of new ones are added. The progressive state of the science has claimed attention, and notices of recent results have been introduced. The Appendix, in addition to the other matter, contains Professor Bessel's late investigation of formulæ for computing Solar Eclipses, Occultations and Transits, reduced to a more elementary form. In the Second Part, the formulæ are applied in the computation of these phenomena. The Tables are nearly the same as in the last edition. Instead, however, of some small ones, which have been omitted, a table of Logarithms, and one of Logarithmic Sines and Tangents to four decimal figures, have been inserted. These are convenient in many computations not requiring greater precision. Haverford School, 9mo. 1842. JOHN GUMMERE. CONTENTS. PAGE CHAP. III. To place an Instrument in the Plane of the Meridian thod of finding the Parallax of a body-Parallaxes and Dis- tances of the Moon and Sun-Their Apparent and Real CHAP. VII. Polar Distance of a Body-Apparent Diurnal Motions of the Fixed Stars, uniform-Motion of the Earth on its Axis CHAP. VIII. Sun's apparent Path-Earth's Annual Motion- Definitions-Positions of the Fixed Stars-Constellations- Catalogues of the Fixed Stars. CHAP. IX. Precession of the Equinoxes-Aberration--Nutation . rent Orbit-Kepler's Laws-Solar Tables-Equation of Time -Sun's Spots, and Rotation on his Axis-Zodiacal Light CHAP. XI. Parallelism of the Earth's Axis-Variations in the Lengths of Day and Night-Seasons-Astronomical Problems CHAP. XII. Definitions-Of the Moon CHAP. XIII. Eclipses of the Sun and Moon-Occultations CHAP. XIV. General Remarks on the Planets-Definitions--Ele- ments of the Orbits of the Planets-Conversion of the Helio- centric Place of a Planet into its Geocentric Place-Retrograde Motion of the Planets-Real Distances, &c., of the Planets . CHAP. XVIII. Classification of the Fixed Stars-Clusters and Nebula-Variable and Temporary Stars-Double Stars- Binary Systems-Proper Motions of some Stars, and Motion CHAP. XXII. Universal Gravitation-Tables of the Elements of the Orbits of the Planets, and of their Masses and Densities. To find the times of longest and shortest Twilight at a given place To find the Annual Variations of a Star in Right Ascension and To find the Aberration of a Star in Right Ascension and Declination To find the Lunar Nutation in Right Ascension and Declination. Given the Eccentricity of the Orbit of a Planet and the Mean PROB. V. The longitudes of two places and the time at one of them being given, to find the corresponding time at the other PROB. VI. To find the sun's longitude, &c., from the tables PROB. VII. To find the sidereal time corresponding to a given PROB. XIII. To determine what Eclipses may be expected to oc- PROB. XIV. To calculate an Eclipse of the Moon PROB. XV. Given the latitude of a place, to find the logarithms of P sin p' and p cos p', in which o' is the geocentric latitude and p the radius of the earth, for the given place . PROB. XVI. To calculate an Eclipse of the Sun for a given place, using the tables of the sun and moon, contained in this work. PROB. XVII. To find a series of places at which an eclipse of the PROB. XVIII. The value of a quantity at three consecutive whole hours, T — 1, T, and T+ 1, being given, to find its value at an intermediate time T', and its hourly variation at that time PROB. XIX. To calculate an Eclipse of the Sun for a given place, |