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I HAD three main objects in the preparation of this Treatise, namely, 1. So to simplify the arrangement of the subject as to give a clear general view, to enable the student to grasp the science as a whole; 2. To save his time by the introduction of shorter and more rapid processes; and, 3. To develop his reasoning powers by constant practice.
I. Simplicity of arrangement. The most inattentive observer can hardly have failed to notice the present confused classification of Arithmetic. Who among the brightest of our scholars can tell us the principles on which it is arranged ? can, at one view, take a clear and definite survey of the whole ground? In the present work, the science has been analyzed into a few simple principles, traced back, indeed, to one grand element, into which all others can be resolved; being, in fact, nothing more than mere abbreviations, by the omission of steps become superfluous by practice.
II. Rapid computation. — The present method of calculating in our schools has been aptly characterized as an awkward mode of spelling figures by taking them singly, in place of reading them in groups. The pupil is so shackled with the multitude of words SUPPOSED to be indispensable, that his progress is necessarily slow and limping. If this work is used agreeably to the directions in p. viii. and elsewhere, these incumbrances will be almost entirely avoided from the outset, and a degree of rapidity quickly attained, which, at first sight, would hardly appear credible. This rapidity of operation, too, is much increased by a very great diminution in the number of figures employed, amounting, in many cases, to more than a half.
III. Intellectual culture. One of the principal objects in the study of the mathematics is the mental culture it affords; and undoubtedly arithmetic cannot be acquired at all without some improvement of the mind. The difference in this respect between the present work and all others is simply this: In the
latter, the pupil reads or commits to memory the reasonings of another mind ; in the former, his rules are the result of his own mental processes. He is led forward by appropriate questions; but he cannot take a single step without active intellectual employment, thus continually eliciting energy of thought, clearness of expression, and fruitfulness of invention.
In the Oral Arithmetic, the lessons should be short, and the questions read slowly, till the pupils become prompt with their answers. Every section should be repeated till it is thoroughly mastered. Answers are occasionally given in the book; but this is merely to indicate the form of expression. None of the exercises will be found too difficult when taken in regular order. Obstacles can only arise from omissions.
In the Written Arithmetic, one question is generally answered by the adjoining one.
At other times the answer is given, unless rendered unnecessary by the mode of proof being pointed out. The exemplifications may be studied on the slate, when the teacher is unable to find time to use the black-board; but such a course is by no means to be recommended. A few of the illustrations may appear obscure when read unconnected with the computations. But all such obscurities will vanish when they are read in their proper connection.
The paragraphs within brackets, [ ], are intended chiefly for the teacher.
In p. 199, 1. 16 from the bottom, a question will be found leading to the formation of a formal rule by the pupil. This, or a similar question, can be repeated wherever the teacher may consider such formulas of any importance. But, where the pupil has acquired clear ideas of the principles of Arithmetic, which he cannot fail to do if he studies this book properly, formal rules will rarely, if ever, be found necessary. See P. 203 and 308—14, for another method of forming rules.
When the pupil is at a loss at a computation, the teacher should neither work it out for him, nor directly instruct him how to proceed. He should merely ask one or more leading questions, or refer him to the proper passage in the book, in order to elicit thought, and lead the pupil to rely on his own
See p. viii. for particular instructions as to the method of using the book.
Arithmetic, general view of the subject,
PART 1.- ORAL ARITHMETIC.
Introduction, addressed to teachers,
Description of the improved numeral frame,
INCREASE AND DECREASE OF INTEGERS, OR WHOLE NUMBERS,
Exercises on the numeral frame.
SECTION I. - Increase and Decrease by unity,
SECTION II. Explanatory,
SECTION III. – Increase and Decrease by a small number, without
causing a change in the ty, or tens,
SECTION IV. - Increase and Decrease by a small number, causing
a change in the ty, or tens, .
SECTION V.- Increase and Decrease by a small number, causing
a change in the tens and the hundreds,
SECTION VI. — Increase and Decrease by larger numbers, the units
causing no change in the tens or hundreds, .
SECTION VII. — Practical Questions,
SECTION VII. - Increase and Decrease by large numbers, the units
causing a change in the tens,
SECTION VIII. - Addition Circles, .
SECTION IX. - Increase and Decrease by large numbers, the tens
causing a change in the hundreds,
SECTION X. - Increase and Decrease by large numbers, the units
causing a change in the tens, and the tens in
SECTION XI. -- Miscellaneous,
SECTION XII. - Increase and Decrease by more than one number,
SECTION XIII. - Practical Questions,
SECTION XIV.- Increase and Decrease by equal numbers, or mul-
tiplication and Division,
SECTION XV. — Explanatory,
SECTION XVI. Multiplication of higher numbers,
SECTION XVII. Definitions,
Section XVIII. - Shortened Multiplication, or Multiplication by
SECTION XIX. — Practical Questions,
SECTION XX. — Division of large numbers by a single digit,
SECTION XXI. - Practical Questions,
INCREASE AND DECREASE OF FRACTIONS, OR BROKEN NUMBERS.
SECTION I. - Fractions of numbers exceeding unity,
SECTION II. Practical Questions, .
SECTION III. Fractions of numbers exceeding unity, continued,
SECTION IV. Practical Questions,
SECTION V. Fractions of numbers exceeding unity, continued, 73
SECTION VI. Practical Questions,
SECTION VII. — Fractions of numbers exceeding unity, continued, . 74
Section VIII. — Practical Questions, .
Section IX. — Fractions of numbers exceeding unity, continued, 77
SECTION X. — Practical Questions, .
FRACTIONS OF UNITY.
SECTION I. — First Principles,
SECTION II. - Prime factors, common multiples, and common divisors, 84
Specimen of the form of the table of prime factors,
SECTION III. Fractional change of form,
SECTION IV. Addition and Subtraction of common fractions, . 91
SECTION V.. Contracted Addition and Subtraction of common frac-
tions, usually called Multiplication and Division, 93
SECTION VI. - Practical Questions,
DETERMINATE FRACTIONS, OR FRACTIONS EXPRESSED IN CONCRETE WORDS,
OR BY SIGNS.
SECTION I. - Change of form, . .
PART II. WRITTEN ARITHMETIC.
Introduction. - Addressed to Teachers,
The Numeral Characters, or Elementary Principles of Increase and
Table of Roman Numerals, .
Table of the Arabic Numerals,
Numeration Table, No. 1.,
English Numeration Table (Note),
Numeration Table, No. 2,
First principles of Decimal Arithmetic,
THE SHORTENED PROCESSES OF INCREASE AND DECREASE OF INTEGERS AND
§ 1. Addition,
§ 2. Subtraction,
Subtraction by Addition,
§ 3. Multiplication,
Involution, or Multiplication by Equal Factors,
§ 4. Division,
Evolution, or Division into two or more equal Factors, . 176
Extraction of the Square Root, or Division into two Equal Factors, 176
Extraction of the Cube Root, or Division into three Equal Factors, 180
Multiplication and Division by Easy numbers,
THE SHORTENED PROCESSES OF INCREASE AND DECREASE APPLIED TO COM-
MON FRACTIONS AND DETERMINATE FRACTIONS.
§ 1. Common Fractions,
I. Change of Form,
II. Addition and Subtraction of Common Fractions,
III. Multiplication and Division of Common Fractions,
Rapid and concise methods of computing with Common Fractions, . 213
IV. Involution and Evolution of Common Fractions,
Practical Exercises on Common Fractional Quantities,
§ 2. Determinate Fractions, or Compound Numbers,
Tables of Coin, Weight, and Measure,
a. Change of Form, .
b. Addition, Subtraction, Multiplication, and Division, of Deter-
1. Multiplication of Duodecimals, or Cross Multiplication, 252
PRACTICAL APPLICATIONS OF THE METHODS OF INCREASE AND DECREASE
Mean numbers and values, commonly called Alligation,
Equation of Payments,
PRACTICAL APPLICATION OF ARITHMETIC BY MEANS OF RATIO AND PRO-
Interest, Discount, Commission, Insurance, and Percentage generally,297
Short processes for the calculation of Simple Interest,
CONTRACTED MULTIPLICATION AND DIVISION OF DECIMAL FRACTIONS.
I. Contracted Multiplication,
II. Contracted Division,
I. Progression by Differences,
II. Progression by Ratios,
Compound Interest by Progression,
Permutation, or Change of Arrangement,
Properties of Numbers,
The Elementary Operations of Arithmetic performed wholly by in-
Synopsis of Principles,