This book has gradually grown out of lessons given orally during many years of teaching. These were written out for the author's own use, not for publication. Originality was not sought for, and all explanations and illustrations which could be of service to the pupils were adopted. As time passed on, and no book appeared precisely suited to the wants of these pupils, or of High Schools in general, the author began to entertain the idea that these lessons might in some measure supply the want so extensively felt. In this hope such completeness has been given to the work as a very limited leisure would allow. It has been revised in manuscript by George P. Bond, Esq., of the Cambridge Observatory, to whom the author is also indebted for. superintending its passage through the press.