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more practical subjects. first, and not to anticipate any principles of 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 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 explans tions, etc. As an illustration, notice the definitions of Ratio, Propor. tion, 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 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 ito 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 actual business transactions. Our Bills and Accounts came out of the stores ; our Taxes, Banking, Exchange, etc., have been submitted to and endorsed by chose connected with the business ; several of the problems on Druies are out of the Oristom House; Insurance has been examined by experts in the business ; the subject of Building Associations, for the first time introduced into 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.

tal and the other on Written Arithmetic and hold that these two sub jects should be embraced in one book. To meet this demand I have made & 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 that 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 teacher 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 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 ons which is doing so much to promote the cause of popular education.

EDWARD BROOKS. STATE NORMAL SCHOOL,

May 10, 1877.

TO TEACHERS.

Teachers will notice that the author's Graded Series of arithmetics consists of three parts. Having completed the Primary Arithmetic, or Part I., which is an introductory course in the science for begin. ners, the pupil is prepared to take up the present volume, Part II. Part II. begins at the beginning of arithmetic and extends as far as Percentage. It embraces Numeration and Notation, Fundamenta. and Derivative Operations, Fractions, Decimals and their applications, Denominate Numbers, and Practical Measurements. After completing Part II., the pupil will be prepared to take up

Part III., which begins at Percentage and finishes the course in arithmetic. These two parts are also bound in one volume for those who prefer the book in that form.

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 unlike ; 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.

11. Mental Arithmetic treats of performing arithmetical operations without the aid of written characters.

12. Written Arithmetic treats of performing arithmetical operations with written characters.

13. Arithmetical Language is the method of expressing numbers.

14. Arithmetical Language is of two kinds, Oral and Written. The former is called Numeration and the latter is called Notation.

NOTE.-A number is really the how many of the collection instead of the collection ; but the definition given, which is a modification of Euclid's, is siinpler and sufficiently accurate.

NUMERATION.

15. Numeration is the method of naming numbers, and of reading them when expressed by characters. It is the oral expression of numbers.

16. Since it would require too many words to give each number a separate name, numbers are named according to the following simple principle :

Principle.- We name a few of the first numbers, and then form groups or collections, name these groups, and use the names of the first numbers to number these groups.

17. A single thing is named one : one and one more are gamed two; two and one more, three; three and one more, four; and thus we obtain the simple names,

One, two, three, four, five, six, seven, eight, nine, ten.

18. Now, regarding the collection ten as a single thing, we might count one and ten, two and ten, three and ten, etc., as far as ten and ten, which we would call two tens. By this principle were obtained the following numbers :

Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty.

19. Proceeding in the same way, we would have two tens and one, two tens and two, two tens and three, etc. By this principle were obtained the following numbers :

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