BY JAMES B. THOMSON, LL. D.

AUTHOR OF MENTAL ARITHMETIC; SLATE AND BLACK-BOARD EXERCISES;
ARITHMETICAL ANALYSIS; HIGHER ARITHMETIC;
EDITOR OF DAY'S SCHOOL ALGEBRA;

LEGENDRE'S GEOMETRY, ETC.

NEW STEREOTYPE EDITION, REVISED AND ENLARGED.

NEW YORK:

IVISON, PHINNEY & CO., 48 & 50 WALKER ST.
CHICAGO: S. C. GRIGGS & CO., 39 & 41 LAKE ST.
BOSTON BROWN, TAGGARD & CHASE. PHILADELPHIA: SOWER, BARNES
& CO., AND J. B. LIPPINCOTT & CO. CINCINNATI: MOORE, WILSTACH,
KEYS & CO. SAVANNAH: J. M. COOPER & CO. ST. LOUIS: KEITH
& WOODS. NEW ORLEANS: BLOOMFIELD, STEEL & 00.
DETROIT: F. RAYMOND & CO.

DAY & THOMSON'S MATHEMATICAL SERIES,

FOR SCHOOLS AND ACADEMIES.

I. ARITHMETICAL TABLES. For Primary Schools. Revised and Enlarged.

II. MENTAL ARITHMETIC, or First Lessons in Numbers. For Children. Revised and Enlarged.

III. RUDIMENTS OF ARITHMETIC, or Exercises for the Slate and Black board. For beginners. Revised and Enlarged. IV. EXERCISES IN ARITHMETICAL ANALYSIS, or Higher Mental Arithmetic. Being a Sequel to Thomson's First Lessons in Numbers. For Advanced Classes.

V. PRACTICAL ARITHMETIC—Uniting the Inductive with the Synthetic mode of Instruction; also illustrating the Principles of CANCELLATION. Revised and Enlarged.

VI. KEY TO PRACTICAL ARITHMETIC. Revised and Enlarged
VII. HIGHER ARITHMETIC, or the Science and Application of
Numbers. For Advanced Classes in Schools and Academies.
This work is designed to develop the Philosophy and appli-
cations of Arithmetic.

VIII. KEY TO HIGHER ARITHMETIC. For Teachers.
IX. THOMSON'S DAY'S ALGEBRA : Being a School edition of
DAY'S LARGE ALGEBRA, with many new illustrations,
and the number of examples much increased.

X. KEY TO THOMSON'S DAY'S ALGEBRA. For Teachers.
XI. THOMSON'S LEGENDRE'S GEOMETRY: with Practical
Notes and Illustrations.

XII. ELEMENTS OF TRIGONOMETRY, MENSURATION, AND LOGARITHMS.

XIII. ELEMENTS OF SURVEYING. (Published soon.)

Entered, according to Act of Congress, in the year 1853,
BY JAMES B. THOMSON,

In the Clerk's Office of the Southern District of New York.

PREFACE.

It has been well said, that "whoever shortens the rosto knowledge, lengthens life." The value of a knowledge of Arithmetic is too generally appreciated to require comment. When properly studied, two important ends are attained, viz: discipline of mind, and facility in the application of numbers to business calculations. Neither of these results can be secured, unless the pupil thoroughly understands the principle of every operation he performs. There is no uncertainty in the conclusions of mathematics; there should be no guess-work in its operations.

What then is the cause of so much groping and fruitless effort in this department of education? Why this aimless, mechanical "ciphering," that is so prevalent in our schools?

Many of these evils, it is believed, arise from the practice of requiring beginners to solve problems above their comprehension, and to learn abstract rules without analysing their principles, or explaining the reasons upon which they are based. Taking his slate and pencil, the pupil sits down to the solution of his problem, but soon finds himself involved in an impenetrable maze. He anxiously asks for light, and is directed "to learn the rule." He does this to the letter, but his mind is still in the dark. By puzzling and repeated trials, he at length finds that certain multiplications and divisions produce the answer in the book; but so far as the reasons of the process, and the principles of the rule are concerned, he is totally ignorant.

It needs no arguments to show that this course is calculated to dampen the ardor of a child, and make him a mechanical