PREFACE.

SCIENCE, in its popular signification, means knowledge reduced to order; that is, knowledge so classified and arranged as to be easily remembered, readily referred to, and advantageously applied. More strictly, it is a knowledge of laws, relations, and principles.

ARITHMETIC is the science of numbers, and the art of applying numbers to all practical purposes. It is the foundation of the exact and mixed sciences, and an accurate knowledge of it is an important element either of a liberal or practical education.

It is the first subject, in a well-arranged course of instruction, to which the reasoning faculties of the mind are applied, and is the guide-book of the mechanic and man of business. It is the first fountain at which the young votary of knowledge drinks the pure waters of intellectual truth.

It has seemed, to the author, of the first importance that this subject should be carefully treated in our Elementary Textbooks. In the hope of contributing something to so desirable an end, he has prepared a series of arithmetical works, embracing four books, entitled, Primary Arithmetic; Intellectual Arithmetic; Practical Arithmetic; and University Arithmeticthe latter of which is the present volume.

PRIMARY ARITHMETIC. This first-book is adapted to the capacities and wants of young children. Sensible objects are employed to illustrate and make familiar the simple combina tions and relations of numbers. Each lesson embraces one combination of numbers, or one set of combinations.

INTELLECTUAL ARITHMETIC. This work is designed to present a thorough analysis of the science of numbers, and to form a complete course of mental arithmetic. I have aimed to make it accessible to young pupils by the simplicity and gradation of its methods, and to adapt it to the wants of advanced students by a scientific arrangement and logical connection, in all the higher processes of arithmetical analysis.

PRACTICAL ARITHMETIC. Great pains have been taken, in the preparation of this book, to combine theory and practice; to explain and illustrate principles, and to apply them to the common business transactions of life-to make it emphatically a practical work. The student is required to demonstrate every principle laid down, by a course of mental reasoning, before deducing a proposition or making a practical application of a rule to examples. He is required to fix and apprehend the unit or base of all numbers, whether integral or fractionalto reason with constant reference to this base, and thus make it the key to the solution of all arithmetical questions. It is hoped, that the language used in the statement of principles, in the definition of terms, and in the explanation of methods, will be found to be clear, exact, brief, and comprehensive.

UNIVERSITY ARITHMETIC. This work is designed to auswer another object. Here, the entire subject is treated as a science. The pupil is supposed to be familiar with the simple operations in the four ground rules, and with the first principles of fractions, these being now taught to small children, either orally or from elementary treatises. This being premised, the language of figures, which are the representatives of numbers, is carefully taught, and the different significations of which the figures themselves are susceptible, depending on the manner in which they are written, are fully explained. It is shown, for example, that the simple numbers in which the value of the unit increases from right to left according to the scale of tens, and the Denominate or Compound numbers in which it increases according to a varying scale, belong to the same class of numbers, and

that both may be treated under the same rules. Hence, the rules for Notation, Addition, Subtraction, Multiplication, and Division, have been so constructed as to apply equally to all numbers. This arrangement, which the author has not seen elsewhere, is deemed an essential improvement in the science of Arithmetic.

In developing the properties of numbers, from their elemen tary to their highest combinations, great labor has been be stowed on classification and arrangement. It has been a lead

ing object to present the entire subject of arithmetic as forming a series of dependent and connected propositions: so that the pupil, while acquiring useful and practical knowledge, may at the same time be introduced to those beautiful methods of reasoning which science alone teaches.

Great care has been taken to demonstrate every proposition -to give a complete analysis of all the methods employed, from the simplest to the most difficult, and to explain fully the reason of every rule. A full analysis of the science of Numbers has developed but one law; viz, the law which connects all the numbers of arithmetic with the unit one, and which points out the relations of these numbers to each other.

In the Appendix, which treats of Units, Weights, and Meas ures, &c., the methods of determining the Arbitrary Unit, as well as the general law which prevails in the formation of numbers, are fully explained. I cannot too earnestly recommend this part of the work to the special attention of Teachers and pupils.

In fine, the attention of Teachers is especially invited to this work, because general methods and general rules are employed to abridge the common arithmetical processes, and to give to them a more scientific and practical character. In the present edition, the matter is presented in a new form; the arrange ment of the subjects is more natural and scientific; the methods have been carefully considered; the illustrations abridged and simplificd; the definitions and rules thoroughly revised and cor

rected; and a very large number and variety of practical ex amples have been added. The subjects of Fractions, Propor

tion, Interest, Percentage, Alligation, Analysis, and Weights and Measures, present many new and valuable features, which are not found in other works.

A Key to the present work has also been published for th ase of such Teachers as may desire it,-prepared with great care, containing not only the answers and solutions of all the examples, but a full and comprehensive analysis of the more difficult ones.

The author has great pleasure in acknowledging the interest which Teachers have manifested in the success of his labors: they have suggested many improvements, both in rules and methods, not only in his elementary, but also in his advanced works. The recitation-room is the final tribunal, and the intelligent teacher the final judge, before which all text-books must stand or fall.

A Key to this volume has been prepared for the use of Teachers only.