and for finding the Least Common Multiple of Fractions; Methods of Equating Accounts; Division of Duodecimals; Exchange, Foreign and Inland; and several important Tables, are among the new features of this edition, which will be found, it is believed, very valuable.

The articles on Money, Weights, Measures, Interest, and Duties are the results of extensive correspondence and much laborious research, and are strictly conformable to present usage, and recent legislation, State and national.

The interpretation of Ratio adopted in this work is the simple and natural method of Chauvenet, Peirce, Loomis, Hackley, Alsop, Day, and other prominent mathematicians of this country, and of nearly all European authorities, including Sir Isaac Newton, Laplace, Legendre, and Bessel.

Questions have been inserted at the bottom of each page, designed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind. It is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions; but vary their form and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject.

The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles.

BENJAMIN GREENLEAF.

The present edition of this work is enriched by a full presentation of the Metric System of Weights and Measures, in accordance with the Tables of Equivalents, established by Congress; and also treats of Government Securities.

Certain articles have been marked by a star (*) to indicate that they may be omitted, at the option of the teacher. BOSTON, May 1, 1873.

ARITHMETIC.

DEFINITIONS.

ARTICLE 1. Quantity is anything that can be measured.

A Unit is a single thing, or one.

A Number is a unit or a collection of units.

An Abstract Number is a number, whose units have no reference to any particular thing or quantity; as two, five, seven.

A Concrete Number is a number, whose units have reference to some particular thing or quantity; as two books, five feet.

The Unit of a Number is one of the same kind as the number; thus, the unit of six is one, and the unit of six pounds is one pound. Arithmetic is the science of numbers, and the art of computing by them.

A Rule is a prescribed mode for performing an operation. The Introductory Processes of arithmetic are Notation, Numeration, Addition, Subtraction, Multiplication, and Division.

The last four are called the fundamental rules, because upon them depend all other arithmetical processes.

NOTATION.

2. Notation is the art of expressing numbers by figures or other symbols.

There are two methods of notation in common use; the Roman and the Arabic.

QUESTIONS. Art. 1. What is quantity? A unit? A number? An abstract number? A concrete number? Arithmetic? A rule? Which are the introductory processes? What are the last four called?-2. What is notation? How many kinds of notation in common use? What are they?

3. The Roman Notation, or that originated by the ancient Romans, employs in expressing numbers seven capital letters, viz.:

I, V, X, L,

C,

D,

M.

one, five, ten, fifty, one hundred, five hundred, one thousand.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations.

1. By a repetition of a letter, the value denoted by the letter is repeated; as, XX represents twenty; CCC, three hundred.

2. By writing a letter denoting a less value before a letter denoting a greater, the difference of their values is represented; as, IV represents four; XL, forty.

3. By writing a letter denoting a less value after a letter denoting a greater, the sum is represented; as, VI represents six; XV, fifteen.

4. A dash (-) placed over a letter makes the value denoted a thousand-fold; as, represents five thousand; IV, four thousand.

V

3. Why is the Roman notation so called? By what are numbers expressed in the Koman notation? What effect has the repetition of a letter? The effect of writing a letter expressing a less value before a letter denoting a greater? Of writing a letter after another denoting a greater value? Ilow many fold is the value denoted by a letter made by placing a dash over it? Repeat the table.