Math 5077.2 B Entered according to Act of Congress, in the year 1870), by ). 3. LIPPINCOTT & CO., In the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. LIPPINCOTT'S PRE88, PHILADELPHIA. r PREFACE. The invention of Analytic Geometry by DESCARTES in the early part of the seventeenth century, quickly followed by that of the Infinitesimal Calculus by NEWTON and LEIBNITZ, produced a complete revolution in the mathematical sciences themselves and accelerated in an astonishing degree the progress of all the sciences in which mathematics are applied, but arrested for a time the progress of pure geometry. The new methods, characterized by great generality and facility in their application to problems of the most varied kinds, offered to the succeeding generations of investigators more inviting fields of research and promises of surer and richer reward than the special and apparently more restricted methods of the ancients. During the eighteenth century hardly any important addition to geometry was made that was not the direct product, either of the Cartesian method alone, or of that method in alliance with the Infinitesimal Calculus. With the present century, however, a new era commenced in pure geometry. The first impulse was given by the Descriptive Geometry of MONGE; then followed CARNOT's Theory of Transversals, PONCELET'S Projective Properties of Figures and Method of Reciprocal Polars, the researches of STEINER, POINSOT, GERGONNE, CAYLEY, MacCULLAGH, and many others, crowned by the brilliant discoveries of CHASLES. All this progress, it is true, has been chiefly in the higher departments of pure geometry, and has not yet essentially changed 3 |