oberts Professor of Mathematics in the New-York State Normal School; Author of "Elementary Arithmetic," "Higher Arithmetic," "Elements of Algebra," "Treatise of Algebra," &c. &c. UTICA: H. H. HAWLEY AND CO. HARTFORD : J. H. MATHER AND CO. QA 453 .P45 Entered according to Act of Congress, in the year 1847, by GEORGE R. PERKINS, in the Clerk's Office of the Northern District of New-York. ALBANY PRINTED BY C. VAN BENTHUYSEN. PREFACE. THERE are two methods of investigating the principles of Geometry. The only method known to the ancients was independent of the aid of Algebra. This method has been so completely developed by EUCLID, as to leave little room for improvement. It is true, modern writers have arrived at many of his conclusions by more simple and concise methods; but, in so doing, they have, in most.instances, sacrificed that rigor of logical demonstration, which so justly constitutes the great merit of his writings. While but little room is left for improving on the model of EUCLID, the modern geometer, by bringing to his aid the principles of Algebra, has greatly enriched the geometry of the ancients, by the discovery of many beautiful relations of magnitudes, which probably would never have been brought to light by the old method. In this work, which is after the model of EUCLID, we have not strictly copied any one author, but have endeavored to select from all the sources within our reach, such parts as we deemed best adapted to our wants. In the solid geometry, or geometry of three dimensions, we have made free use of PETER BARLOW's arrangement, as given in the Encyclopædia Metropolitana; which, indeed, is but a slight modification of LEGENDRE's method. We have found, from experience in teaching, that, as a general thing, beginners in the study of geomety consider it as a dry, uninteresting science. They have but little difficulty in following the |