From the PREFACE . THE following treatise on the Elements of Analytical Geometry and the Differential and Integral Calculus is the result of long experience in educating young men. It is, in fact, the substance of that which the Author has taught in the Free Academy for several years; and one great object in bringing it before the scientific instructors of our country is an endeavor, on his part, to render the student's path, particularly through the Calculus, as smooth, and plain, and easy as the nature of that science will admit. The Author well knows that a correct system of ' education requires constant mental effort on the part of the student who aspires to be a thorough scholar, and that a subject which is simplified too much in a text-book becomes unfitted for the object for which it was intended-namely, to enlarge and strengthen the powers of the mind. But he thinks that this fault can scarcely occur in any treatise on the "Theory of the Variation of Variables and their Functions," a science which, under the most favorable light that can be thrown upon it, is sufficiently obscure for the mind, at the early age in which the young men of this country begin the study of it. It will be seen that the Author has used the old symbols exclusively, and that he has confined his system to the Method of Limits. He has done this, not on account of any partiality he feels for that system, for the Infinitesimal Method generally reaches the conclusions in a far less time and a much shorter space, but solely from his belief that the former is better adapted to the great purpose-of teaching, and that it has a greater tendency to draw out the energies of the stu dent, expand his genius, invigorate his thoughts, and impart a keener perception to his intellectual powers. In discussing the Ellipse and Hyperbola together, the Author has followed the example of several eminent writers on Analytics. In this, as also in the method he has adopted in the discussion of the circle and the transformation of co-ordinates, and elsewhere, he has been enabled to reduce the Treatise to its minimum size without the omission of a single essential proposition. In the recitation-room, perhaps it would be better for one student to take the proposition in reference to the ellipse, while another student takes the same for the hyperbola; in this way the number of examples becomes doubled. With these prefatory remarks, the volume is respectfully submitted to the learned and scientific gentlemen who are engaged in teaching those useful and important branches of pure Mathematics. They will find it precisely what it purports to be-nothing more and nothing less - a Treatise on the Elements of Analytics and the Differential and Integral Calculus, prepared at the request, and for the use of the students in the New York Free Academy, but equally adapted to the instruction given in our colleges and other seminaries of learning.