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In Books I and II of this series, classification is made subordinate to gradation. In Book III, it is assumed that the pupil has a body of knowledge sufficient to enable him to profit by a topical arrangement of subjects. Classification and generalization are therefore the features of this book.
In answer to what is believed to be a general demand, the number of topics has been greatly reduced. The fundamental operations in their application to simple numbers, decimals, United States money, denominate numbers, and literal quantities, are treated briefly under the four general heads, Addition, Subtraction, Multiplication, and Division. The other topics presented are,- Properties of Numbers, Divisibility of Numbers, Fractions, Percentage and Its Applications, Ratio and Proportion, Powers and Roots, The Metric System, and Denominate Numbers.
Each page of the book is a unit of the greater ten-page unit. Upon the first six pages of every ten pages, a part or the whole of some general topic is presented. Upon the seventh and eighth pages, the algebraic phase of this topic appears. The ninth page is devoted to elementary work in geometry, and the tenth to miscellaneous problems. This arrangement makes the book convenient for reference and review, and will aid the pupil in gaining a perception of the relation of the topics and sub-topics.
The introduction of the elements of algebra and geometry will greatly increase the interest of all the pupils, will prove invaluable to those who leave school at the end of the eighth year, and become a helpful stepping stone to such as take a high school course.
The author is under great obligation to Supt. J. C. Burns of Monmouth, Illinois, whose criticism of the manuscript of Book III has greatly improved the form of expression, and to Dr. J. B. Shaw, Professor of Mathematics in Illinois College, to whom many queries have been submitted, the answers to which have helped to bring the book into harmony with the best mathematical thought of the day. In justice to these scholars, it should be said that neither of them is responsible for any errors that may be found in the book.
For all that is new and peculiar on pages 211 and 213, the author is indebted to Mr. Geo. R. Parker, a blind man who has taught mathematics in The Illinois Institution for the Education of the Blind with marked success for many years.
F. H. H. JACKSONVILLE, ILLINOIS, June, 1898.
NOTE.-The following brief statements concerning the elementary branches of mathematics, and the place of arithmetic in mathematical science, are designed for the teacher rather than for the pupil. After the completion of the book, these statements should be read aloud in class and the pupil encouraged to comment upon them and to discuss their meaning. After this has been done, the pupils may be required to commit to memory such parts as seem most important.
1. Mathematics treats of the measurement and comparison of magnitudes.
Geometry, arithmetic, and algebra are the elementary branches of mathematics.
2. Geometry treats of the measurement and comparison of certain magnitudes known as lines, angles, surfaces, and solids.
Measurement makes number necessary.
Primarily, number signifies ratio. The number six suggests that some magnitude, A, is six times some other magnitude, B; and, con. versely, that the magnitude, B, is one sixth of the magnitude, A.
Number is ratio.
Secondarily, number signifies aggregation. The number six suggests the aggregation of six minor magnitudes into one group, or major magnitude.
NOTE.— It is the secondary aspect of number that the child first apprehends. Thus, that which is logically secondary is pedagogically primary. To the child, the number six suggests a group of six (marbles, apples, inches). It is undoubtedly true that the mature mind ordinarily conceives of number in its aggregative aspect. It is the mathematician only who habitually thinks of number as ratio.
3. Arithmetic and Algebra treat of measured magnitudes and their relation-hence of number.
Every arithmetical or algebraic problem is one of the following two simple problems, or may be resolved into such as these: