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PREFAOE.

EDUCATION is progressive. The developu.ext 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 pro fession.

This advance in education has been nownere more noticeable than in the improvements of text-books upon Arithnietic. A few years ago an un pretending little work, Colburn's Intellectual Arithmetic, was pre sented 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 commcn 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 opon 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 business relations of life.

ARRANGEMENT.-Tke arrangement of the work is believed to be strictly logical and at the same time, practical, being adapted to the natural mental growth as.d development of the pupil. The mottoes have been,

from the easy to the difficult, from the simple to the complex, from the browon 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 Proponion, Equation of Payments after Proportion, and other arrangements have been determined by the same principle.

THE REASONING.-All reasoning is comparison. A comparison re quires 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 explana tions, 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 expressen 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 prolis, 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 briet 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 aritbmetical machines.

APPLICATIONS.—One of the most prominent features of the work is its practical character. The applications of the science are not the thought of the scholar as what business may be, but represent the actual business of the day. Many of the problems and processes are derived from actuar business transactions. Our Bills and Accounts came out of the stores; our 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 Orstom House; Insurance has been examined by experts in the business; the subject of Building Associations, for the first time intro duced inw an arithmetic, was partly prepared by one practically famil. iar with the subject; etc.

UNION OF MENTAL AND WRITTEN.—Another leading feature of the work is the union of mental and written arithmetic in one book. Many who recognize the importance of Mental Arithmetic think that it takes too much time for the pupil to study two separate books—one on Men

mal and the other on Written Arithmetic- and hold that these two subjects should be embraced in one book. To meet this demand I have made a complete and harmonious combination of the two subjects, introducing many of those forms of analysis that have given such popularity to my Mental Arithmetic. It is this combination what gives the work its name, The Union Arithmetic; and this union will be found to be not a mere nominal thing, but a reality. In the study of the work the pupil can obtain quite a thorough course in arithmetical analysis while he is becoming familiar with the art of computation and the application of the art to business. These mental exercises are so arranged and printed that any vescher who prefers to omit them can do so without any inconvenience to either the pupil or the teacher.

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 Pro portion 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 a.iso 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, 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.

EDWARD BROOKS. STATI NORYAL SOROOL,

May 10, 1877.

NOTE TO TEACHERS.

AMPLES.

The attention of teachers is respectfully invited to the arrange ment of the present edition of this work for SUPPLEMENTARY Ex.

This is a new feature in arithmetic, and one that will, I doubt not, meet the approval of intelligent teachers.

These supplementary examples can be used as the best interests of the pupils require. They will be especially valuable in furnishing new matter for review, or in the final review of the work. 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 bringing of Insurance earlier in the course, and the placing of Partitive Proportion, Conjoined Proportion, and Medial Proportion later in the work under the head of Supplement, is regarded as an improvement. The present edition thus represents the latest and most approved methods of business calculation, as well as the most advanced arithmetical thought of the day.

EDWARD BROOKS. PHILADELPHIA, June 20, 1888.

THE

NORMAL

UNION ARITHMETIC

SECTION I.

ARITHMETICAL LANGUAGE.

1. Arithmetic is the science of numbers and the art of computing with them.

2. A Unit is a single thing or one. A thing is a concrete unit; one is an abstract unit.

3. A Number is a unit or a collection of units. Numbers are concrete and abstract.

4. A Concrete Number is one in which the kind of unit is named; as, two yards, five books.

5. An Abstract Number is one in which the kind of unit is not named; as, two, four, etc.

6. Similar Numbers are those in which the units are alike; as, two boys and four boys.

7. Dissimilar Numbers are those in which the units are anlike ; as, two boys and four books.

8. A Problem is a question requiring some unknown result from that which is known.

9. A Solution of a problem is a process of obtaining the required result.

10. A Rule is a statement of the method of solving a problem.

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