PREFACE.

EDUCATION is progressive. The development of the popular mind is becoming the transcendent question of the day. Improvements are being made in every department, dull routine is giving way to intellectual activity, instruction is becoming a science, and teaching a profession.

This advance in education has been nowhere more noticeable than in the improvements of text-books upon Arithmetic. A few years ago an unpretending little work, Colburn's Intellectual Arithmetic, was presented to the public. That little work touched Arithmetic as with the wand of an enchantress, and transformed it from a dry collection of mechanical processes to a thing of interest and beauty. It laid the foundation of that system of Mental Arithmetic which has infused a new spirit into the science of numbers, and has done more than any other influence to vitalize the methods of common school instruction in this country.

In presenting a new work upon the subject, I desire to acknowledge my obligations to this and other works which have followed it. Bringing to the task the reflection and experience of many years of educational labor, I hope to be able to present a text-book upon Arithmetic which will take an honorable position among the many valuable works upon the subject which are doing so much for the educational interests of the country. Some of the general and special features of this work will be briefly noticed.

METHOD OF TREATMENT.-The method of treatment is both Induc tive and Deductive, embracing Analysis and Synthesis. In some cases both of these methods are employed in the development of the same subject; in other cases they are combined in the same solution or explanation, and such combination is characteristic of the entire work. I have endeavored to meet the wants of both teacher and pupil, by preparing a work convenient for instruction, adapted to the natural and logical development of the mind of the pupil in the study of numbers, and containing such applications as will prepare students for the busi rese relations of life.

ARRANGEMENT. The arrangement of the work is believed to bo strictly logical and, at the same time, practical, being adapted to the natural mental growth and development of the pupil. The mottoes have been-from the easy to the difficult, from the simple to the complex, from the known to the unknown. Care has been taken to present the simpler and more practical subjects first, and not to anticipate any principles or processes before the pupil is prepared for them. Thus, I have placed Compound Numbers after Fractions, Percentage before Ratio and Proportion, Equation of Payments after Proportion, and other arrangements have been determined by the same principle.

THE REASONING.-All reasoning is comparison. A comparison requires a standard, and this standard is the fixed, the axiomatic, the known. The law of correct reasoning, therefore, is to compare the complex with the simple, the theoretic with the axiomatic, the unknown with the known. This law is kept prominently before the mind in the development of this work, and upon it are based its definitions, solutions and explanations, etc. As an illustration, notice the definitions of Ratio, Proportion, etc., the method of stating a proportion, etc.

SOLUTIONS.-The solutions and demonstrations are so simple and clear, that they may be understood by very young pupils, yet they are expressed in language concise and logically accurate, and in the form which the pupil should be required to use at recitation. A solution may be too concise to be readily understood, and it may also be too prolix, the idea being smothered or concealed in a multiplicity of words. Both of these errors I have endeavored to avoid, remembering that the highest science is the greatest simplicity.

RULES.-The rules or methods of operation are expressed in brief and simple language, and are given as the results of solutions and explanations. I have endeavored to lead the pupil to see the reason for the different processes, thus enabling him to derive his own method of operation based upon such reasoning. The object has been to develop mind as well as the power of computation-to make thinkers rather than arithmetical machines.

APPLICATIONS.—One of the most prominent features of the work is its practical character. The applications of the science are not the idea of the scholar as to what business may be, but represent the actual business of the day. Many of the problems and processes are derived from actual business transactions. The Bills and Accounts came out of the stores; the Taxes, Banking, Exchange, etc., have been submitted to and endorsed by those connected with the business; several of the problems on Duties are out of the Custom House; Insurance has been examined by experts in the business; the subject of Building Associations, for the first time introduced into an arithmetic, was mainly prepared by one practically famil iar with the subject; etc.

SPECIAL FEATURES.-There are several special features peculiar to this work, to which we desire to call attention.

1st. Many new definitions, as of Fraction, Least Common Multiple, Percentage, Ratio, etc.

2d. New and concise method of explaining Greatest Common Divisor, and a method of Least Common Multiple not usually given.

3d. The two distinct methods of the development of Fractions, the relation of fractions, the method of stating a problem in Simple Proportion and reason for it, and the development of Compound Proportion.

4th. The Analytic and Synthetic methods of developing Involution and Evolution, the greater attention to Involution as a preparation to Evolution, a new method of cube root, etc.

5. Great number and variety of problems, especially after the Fundamental Rules, Fractions, etc., and at the close of the book. Other features also important, will present themselves upon a careful examination.

It should be stated that this work was first published in 1863, and that some of the definitions and processes which were then new have since been introduced into other works. The present edition is thoroughly revised, and brought up to the very latest methods of business calculations.

Thanking my friends for the cordial reception given to my previous labors, I send forth this new volume, with the earnest desire that it may meet their approbation, and aid in the development and diffusion of a deeper interest in the beautiful science of numbers-a science which practically lies at the foundation of all science and all thought, and one which is doing so much to promote the cause of popular education.

STATE NORMAL SCHOOL,

May 10, 1877.

EDWARD BROOKS.

THE attention of teachers is respectfully invited to the arrangement of the present edition of this work for SUPPLEMENTARY EXAMPLES. This is a new feature in arithmetic, and one that we believe will meet the approval of intelligent teachers.

These Supplementary Examples can be used wholly or partly, as may be thought best for the interests of the pupils. They will be especially valuable in furnishing new matter for reviews or in the final review of the book. Classes whose time for the study of arithmetic is somewhat limited may omit them and still have a complete and comprehensive course in this branch.

A few other changes have been made in the present edition, rendered necessary by recent changes in several lines of business. The work thus represents the latest and most approved methods of business calculation as well as the most advanced arithmetical thought of the day.

PHILADELPHIA, June 16, 1888.

EDWARD BROOKS.