THE

NORMAL

UNION ARITHMETIC

GRADED COURSE.

PART II.

BY

EDWARD BROOKS, A. M., Pí. D.,

PRINCIPAL AND PROFESSOR OF MATHEMATICS IN PENNSYLVANIA STATE NORMAL
SCHOOL, AND AUTHOR OF "THE NORMAL SERIES OF ARITHMETICS," "NORMAL
ELEMENTARY ALGEBRA," "NORMAL GEOMETRY AND TRIGONOMETRY,"
"PHILOSOPHY OF ARITHMETIC," ETC.

"Analysis and Synthesis -the magic wand whose touch unfolds the mysterious and
beautiful relations of numbers."

PHILADELPHIA:

CHRISTOPHER SOWER COMPANY,

614 ARCH STREET.

CONTENTS.

90 DENOMINATE FRACTIONS.

Reduction,

91

92

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 nownere 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 Inductive 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.-The arrangement of the work is believed to be 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