METHOD OF TREATMENT.-The method of treatment is both Inductive and Deductive, embracing Analysis and Synthesis. In some cases both of these methods are employed in the development of the same subject, as in Fractions, Percentage, Involution, and Evolution. In other cases they are combined in the same solution or explanation, and such combination, though not always obvious, 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, and adapted to the natural and logical development of the mind of the pupil in the study of numbers. ARRANGEMENT.-The arrangement of the work is believed to be strictly logical, and, at the same time, practical, being adapted to the mental growth of the pupil. The motto has been, from the easy to the difficult, from the simple to the complex. The object has been to present the simpler and more practical subjects first, being careful not to anticipate any principles or processes. Thus, I have placed United States Currency before, and Compound Numbers after, Fractions, 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 to the simple, the theoretic to the axiomatic, the unknown to the known. This law is kept prominently before the mind in the development of this work, and upon it are based its solutions and explanations, its definitions of Ratio, Proportion, etc., its method of stating a proportion, etc. SOLUTIONS.-The solutions and demonstrations are simple and clear, that they may be understood by very young pupils, but 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. SPECIAL FEATURES.-There are several special features peculiar to this work, to which we desire to call attention. 1st. New definitions of Number, Fraction, Least Common Multiple, Ratio, etc. 2d. New and concise method of explaining Greatest Common Divisor, and a method of Least Common Multiple not usually given. 3d. The development of Fractions by two distinct methods, the relation of fractions, and the simple and concise methods of explaining Greatest Common Divisor and Least Common Multiple of fractions. 4th. The method of stating a proportion in Single Rule of Three and reason for it. The development of Compound Proportion. 5th. The Analytic and Synthetic methods of developing Involution and Evolution, and the greater attention to Involution as a preparation to Evolution. 6th. 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. Thanking my friends for the cordial reception given to my previous labors, I send forth this little 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, August 6th, 1863. EDWARD BROOKS. THE NORMAL WRITTEN ARITHMETIC. SECTION I. ARTICLE 1. The first and simplest idea of Arithmetic is the Unit or One. 2. A collection of units gives rise to numbers; from the combination and comparison of which arises the subject of Arithmetic. 3. Arithmetic is the science of numbers and the art of computation. 4. Science is principles systematized. Art is the application of the principles of science. 5. A Unit is a single thing. Number is the how many of collection of units. a 6. A Number is how often a unit is repeated or is contained in a collection. 7. A Concrete Number is one which refers to some particular unit; as, two yards, five books. 8. An Abstract Number is one which does not refer to any particular unit; as, two, five. 9. Similar Numbers are those in which the units are the same; as, two boys, four boys. Dissimilar Numbers are those in which the units are different; as, two boys, four roses. 10. When Arithmetic treats alone of the properties, relations, etc., of abstract numbers, it is called Abstract, Theoretical, or Pure Arithmetic. 11. When the numerical operations are with concrete numbers, it is called Concrete or Practical Arithmetic. 12. When the operations are performed in the mind, without the aid of written characters, it is called Mental Arithmetic ; when with written characters, Written Arithmetic. ARITHMETICAL LANGUAGE. 13. There are two kinds of Arithmetical Language,—the Oral, or Spoken, and the Written. The former is called Numeration, the latter Notation. 14. Numeration is the art of naming numbers and of reading them when expressed by characters. PRINCIPLE.—The principle of Numeration is to name a few numbers and then form groups or collections, and use the simple names to number these groups. Naming numbers in this way a single thing is one; one and one more are two; two and one more, three; three and one more, four; and in a similar manner we have five, six, seven, eight, nine, ten. Regarding the collection ten as a single thing, we have one and ten, two and ten, etc., up to ten and ten, which we call two tens Proceeding in the same way we have two tens and one, two tens and two, etc., to ten tens, which gives a new group, called hundred. In this manner any number, however large, may be easily named, and we have a language simple, beautiful, and convenient. Some of these names have become greatly changed by custom, so that with small numbers we can hardly perceive the principle of naming. Instead of one and ten we use eleven, meaning one left after ten, and in place of two and ten we say twelve, meaning two left after ten. Omitting the "and," changing ten to teen, three to thir, etc., we have thirteen, fourteen, etc. Changing teen to ty, two to twen, from the Saxon twain, etc., we have twenty, thirty, forty, etc. The next group consists of ten hundreds, which is called Thousand, the next group consists of a thousand thousands, which is called Million, the next group consists of a thousand millions, which is called Billion, Having seen how numbers are named, we are prepared to learn how to write them. 15. Notation is the method of writing numbers. There are three methods of writing numbers :— 1st. By words, or common language. |