JOHN F. STODDARD, A.M, AUTHOR OF "STODDARD'S ARITHMETICAL SERIES," ETO. PROFESSOR AND W. D. HENKLE, OF MATHEMATICS, IN THE SOUTH-WESTERN NORMAL NEW YORK: PUBLISHED BY SHELDON & CO., 115 NASSAU STREET. Entered, according to Act of Congress, in the year 1859, by SHELDON AND COMPANY. In the Clerk's Office of the District Court for the Southern District of New York. STEREOTYPED BY THOMAS B. SMITH, 82 & 84 Beekman-st., N. Y. PREFACE. THE former edition of this volume met with a sale which far exceeded the expectation of its authors, as the work was written only for good teachers and good schools. The original design of writing an additional volume has been abandoned, and, in the present edition, chapters have been added on Permutations and Combinations, Calculus of Probabilities, Logarithms, Interest and Annuities, Indeterminate Analysis, Diophantine Analysis, and the General Theory of Equations, including Transformation of Equations, Limits of Roots, Equal Roots, Integral Roots, Sturm's Theorem, Horner's Method of Resolving Numerical Equations of all Degrees, and Cardan's Formula for Cubic Equations. This volume is, perhaps, the most extensive treatise on Algebra which has ever been published in America, and this fact leads us to hope that it will be extensively used in the higher grade of Institutions where the mathematical sciences receive considerable attention. Although an unusual number of practical examples have been inserted in chis volume, on the principle that algebraic skill can only be acquired by extensive practice, yet the teacher should select only such as may be best adapted to the particular class which he has under his instruction. While some classes should solve all the examples, others should turn their attention only to the most difficult, and others, again, should omit the most difficult examples. The judicious teacher, who becomes familiar with the work, can select such subjects as may be best suited to his class. The arrangement of this work is, in many respects, new. The plan of treating every subject as the solution of a general problem, or as the demonstration of a theorem, it is hoped, will greatly facilitate rigid class instruction. Many of the demonstrations have been rendered so clear, that they may be readily comprehended by students of ordinary capacity; others, which are of a more abstruse character, will require close application. The attention of teachers is called to the classification of Algebraic Symbols in Chap. I.; the explanation of subtraction, and Articles (81), (82), (83), (84), (94), (95), (96), (97), in Chap. II.; Articles (112), (113), and (114) in Chap. III.; the demonstration of the rule for finding the Greatest Common Divisor of two polynomials in Chap. IV.; Articles (160) and (162) in Chap. VI.; the general Discussion of the Courier Problem in Chap. X.; and the Demonstration of the Multinomial Theorem in Chap. XVIII. The method of solving equations of the third and the fourth degree as set forth in Articles (335), (337), (354), and (355), although tentative in its character and not practically general, furnishes the means of resolving many problems which have heretofore been considered difficult. This method is considered valuable in an educational rather than in a scientific point of view. The Chapter on "Continued Fractions" will be found to be a fuller discussion of this interesting subject than has yet been given in any other treatise published in this country. We might call attention to other parts of the work, but it is deemed unnecessary. This volume is now submitted to an intelligent public, with the hope that after a careful examination it will meet with the approbation of all those who favor the use of text-books that do not attempt to simplify by the omission of that which is difficult. July 4, 1859. J. F. S. AND W. D. H. [NOTE.-A KEY to this volume will be issued without delay.] |