This volume is the outgrowth of a course of lectures given annually by the author at the University of Chicago during the last six years. These lectures have been open to senior college students and to graduate students who have not had the equivalent of this work. They have been taken by students of Astronomy, by many making Mathematics their major work, and by some who, though specializing in quite distinct lines, have desired to get an idea of the processes by means of which astronomers interpret and predict celestial phenomena. Thus they have served to give many an idea of the methods of investigation and the results attained in Celestial Mechanics, and have prepared some for a detailed study extending into the various branches of modern investigations. The object of the work, the subjects covered, and the methods of treatment seem to have been amply justified by this experience.