RAY'S ALGEBRA PART SECOND: AN ANALYTICAL TREATISE, DESIGNED FOR HIGH SCHOOLS AND COLLEGES. m BY JOSEPH RAY, M. D. Stereotype Edition. PUBLIC PUBLISHERS: W. B. SMITH & CO., CINCINNATI OG F MATHEMATICAL COURSE. NEW YORK IC LIBRARY. ARITHMETIC. TOR, RAY'S ARITHMETIC, PART FIRST; very simple lessons for little RDEN FOlearners.s. RAY'S ARITHMETIC, PART SECOND; a complete text-book in RAY'S ARITHMETIC, PART THIRD; for schools and academies ; ALGEBRA. RAY'S ALGEBRA, PART FIRST; for common schools and acade mies; a simple, progressive, and thorough elementary treatise. RAY'S ALGEBRA, PART SECOND; An Analytical Treatise, embracing Sturm's Theorem, and Horner's Method of Approximation. KEY TO RAY'S ALGEBRA, PARTS FIRST and SECOND; Complete in one volume, 12 mo. PREPARING FOR PUBLICATION, ELEMENTS OF GEOMETRY; embracing Plane and Solid Geometry, with numerous interesting and practical exercises. TRIGONOMETRY AND MENSURATION; containing Logarithmic Computations, Plane and Spherical Trigonometry, with their applications, and Mensuration of Planes and Solids, with Logarithmic and other Tables. SURVEYING AND NAVIGATION; containing Surveying and Levelimg: Navigation; Barometric hights, &c. The above will be followed by works on Analytical Geometry:-. Mechanical Philosophy:- Differential and Integral Calculus:-and Mathematical Astronomy. PROFESSOR Kay, during a long period devoted to actual instruction in these several branches, has prepared much of the material requisite for the respective volumes. They will appear as rapidly as a due regard to their careful publication will permit. Entered according to act of Congress, in the year Eighteen Hundred and FiftyTwo, by WINTHROP B. SMITH, in the Clerk's Office of the District Court of the United States, for the District of Ohio. PREFACE. ALGEBRA is justly regarded one of the most interesting and useful branches of education, and an acquaintance with it is now sought by all who progress beyond the more common elements. To those who would know Mathematics, a knowledge not merely of its elementary principles, but also of its higher parts, is essential; while no one can lay claim to that discipline of mind which education confers, who is not familiar with the logic of Algebra. It is both a demonstrative and a practical science - a system of truths and reasoning, from which is derived a collection of Rules that may be used in the solution of an endless variety of problems, not only interesting to the student, but many of which are of the highest possible utility in the arts of life. The object of the present treatise is to present an outline of this science in a brief, clear, and practical form. The aim throughout has been to demonstrate every principle, and to furnish the student the means of understanding clearly the rationale of every process he is required to perform. No effort has been made to simplify subjects by omitting that which is difficult, but rather to present them in such a light as to render. their acquisition within the reach of all who will take the pains' to study. To fix the principles in the mind of the student, and to show their bearing and utility, great attention has been paid to the preparation of practical exercises. These are intended rather to illustrate the principles of the science, than as difficult problems to torture the ingenuity of the learner, or amuse the already skillful Algebraist. An effort has been made throughout the work, to observe a natural and strictly logical connection between the different parts, so that the learner may not be required to rely on a principle, or |