tation of the fractional and negative exponents is exhibited as a necessary consequence of the definition. The demonstration of the Binomial Theorem for negative and fractional exponents (§§ 291-294), and the development of the fundamental logarithmic formula (§§ 320–323) are substantially those of Lagrange. The nature of the Modulus (§§ 327-332), and some of the properties of logarithmic differences (§§ 333-336) are discussed more fully than I have seen them in any elementary treatise. Familiarity with these principles is of great advantage to the student, and their discussion is, by no means, difficult. A table of the principal formulæ of the book is placed after the table of contents, for convenience of reference and review. It has also the advantage of generalizing, and bringing into one view, principles exhibited, with more or less fulness, in different parts of the book. For the suggestion of this table, I am indebted to Mr. Richards, the able Principal of Kimball Union Academy. I am also very greatly indebted to my associates, Professors Crosby and Young, for valuable suggestions and criticisms. In correcting the proofs of the last half of the work, I have had the assistance of Mr. Edward Webster, a recent graduate of the College, whose tastes and attainments qualify him to do excellent service in the cause of science. S. C. Dartmouth College, May 1, 1849. 1. Multiplication,-Monomials,-Signs,-Degree, IV. Polynomials,-Detached coefficients, Polynomials,-Detached coefficients, Synthetic division,-Infinite series, Theorems, (a + b)2, (a+b)(a—b), |