Q A TREATISE ON ALGEBRA. BY ELIAS LOOMIS, A.M., PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN THE UNIVERSITY OF THE OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES, &c. NEW YORK: HARPER & BROTHERS, PUBLISHERS, 82 CLIFF STREET. 1846. Entered, according to Act of Congress, in the year 1846, by ELIAS LOOMIS, In the Clerk's Office of the Court of the Southern District of New York. ΤΟ THE COUNCIL OF THE UNIVERSITY OF THE CITY OF NEW YORK, THIS TREATISE Es respectfully Enscribed, BY THEIR OBEDIENT SERVANT, THE AUTHOR. PREFACE. THE present Treatise is designed to supply a deficiency which has long been felt. Having been engaged for many years in imparting instruction in Algebra, I have found no text-book entirely adapted to the wants of my pupils. Some treatises, from their extreme conciseness, and their aiming at the utmost rigor of demonstration, are too difficult for the majority of students, while others are far behind the present state of science. The present Treatise was specially designed for the use of the students of New York University, but it is believed to be adapted to the wants of students generally in American colleges. It is designed for youth of fifteen or sixteen years of age, who are supposed to possess ordinary abilities and aptitude for study. I have not, therefore, attempted to demonstrate every principle in the most general and rigorous manner. Such demonstrations would often be unintelligible and repulsive to the majority of students. Nor, on the contrary, have I contented myself with mere mechanical rules. I have aimed to follow the natural order of the youthful mind, deducing general principles from particular examples, subjoining afterward a complete demonstration, whenever it was thought that its force would be clearly apprehended. I have aimed to lead the student to generalize every principle. For this purpose, nearly every problem is twice stated; first in a restricted form, and afterward in a more general one; and I have dwelt upon the interpretation of some peculiar |