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PREFACE.

IT is more than four years since this work was published. During the whole of this time it has been in constant use under my own superintendence; and, consequently, I have had opportunity to ascertain what were its defects, and wherein a difference of arrangement, or other modifications, would be desirable. I have, also, consulted experienced teachers with direct reference to the present revision of the work, and now submit the result to the public.

I am confident that great improvements will be found in the following particulars. In the statements of properties, relations, and principles-in the phraseology of definitions and of Rules-in the methods of illustration-in the order of arrangement of the subjects treated; indeed, throughout the entire work.

My object has been to be concise, yet lucid; to reach the radical relations of numbers; and to present fundamental principles in analyses and examples, that shall leave nothing obscure, yet that shall not embarrass by multiplied processes, or enfeeble by ininute details. I hold to the idea that a suffi. ciency of illustration to lay open thoroughly the subject treated, is all that is desired; and that whatever is redundant impairs the force of what is essential. Both teachers and pupils will, as I judge, be

ber-fitted by thus leaving them somewhat to the ac n of their own minds.

It is not easy for me to specify points to which attention may be directed. But I would suggest, the definition of the values of Figures-of the Zero; the illustration of Subtraction; the general treatment of Vulgar Fractions; the introduction of Decimal Fractions before Federal Money and of Duodecimals immediately after Denominate Decimals; the whole arrangement of Percentage and Interest; the method of finding the Cash Balance in Equation of Payments. And last, but not least, the method of extracting the Cube Root, by means of auxiliary columns. To this method I ask the attention of teachers generally. I believe I have omitted no step necessary to make it perfectly intelligible; and for conciseness and beauty, as well as for practical use, it is incomparably superior to the usual method.

Throughout the entire work many new examples have been given, which have been formed with much care, having the different parts so related as to bring out, when solved, exactly the principle designed. Many of these questions contain statistical and historical facts which it is desirable for all to know, thus giving an interest to the questions which they could not possess in an abstract and simply numerical form.

UTCA, March, 1849.

GEO. R. PERKINS.

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ARITHMETIC.

ARTICLE 1. ARITHMETIC is the science of numbers. The operations of arithmetic are performed by the aid of five distinct rules, viz.: Numeration, Addition, Subtraction, Multiplication, and Division. These are usually called the FUNDAMENTAL RULES of arithmetic, because all other niles are founded upon them.

What is Arithmetic? How many distinct rules has it for its operations? Repeat their names. What are these usually called? Why are they so called?

NUMERATION.

2. NUMERATION explains the method of reading written numbers.

Notation is the writing down of numbers.

Various methods of notation and numeration were used by the ancients. We shall content ourselves with mentioning two, the common or Arabic method, and the Roman method.

In the common method ten characters are employed. These characters when written are,

, 2, 3, 4, 5, 6, 7, 8, 9, 0

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