subjects being uniform and essentially as follows: 1o. All necessary definitions are given; 2°. A few mental exercises are then introduced, being so worded and so arranged as to lead the pupil to a knowledge of the fundamental principles of the subject under consideration; 3°. The principles thus developed are used in demonstrating the required rule; and 4°, The rule is then illustrated and enforced by a sufficient number of graded examples and problems. The Author takes this opportunity to thank the many teachers who have aided him by valuable suggestions and criticisms. COLUMBIA COLLEGE, 1. A Unit is a single thing; as, one pound, one foot, one day. 2. A Number is a unit, or a collection of units; as, one pound, three days, five feet. 3. Arithmetic is the science of numbers. It treats of the properties and relations of numbers, and of the methods of computation by means of numbers. NOTE.-In what follows, the expressions 1°, 2°, 3°, etc., are read first, second, third, etc. FORMATION OF NUMBERS. 4.-1°. Numbers from one to ten are formed by collect ing simple units, or ones. A single unit is called one; one and one more are two; two and one more are three; three and one more are four; and so on, to ten. 2o. Numbers between ten and one hundred are formed by collecting tens and ones. One ten and one are eleven; one ten and two are twelve; one ten and three are thirteen; and so on to two tens, or twenty. Two tens and one are twenty-one; two tens and two are twenty-two; and so on to three tens or thirty. Four tens are forty; five tens are fifty; and so on to ten tens, or one hundred. The intermediate numbers between thirty and forty, forty and fifty, and so on, are formed in the same manner as those be tween twenty and thirty. 3o. Numbers between one hundred and one thousand are formed by collecting hundreds, tens, and ones; numbers between one thousand and ten thousand are formed by collecting thousands, hundreds, tens, and ones; and so on, indefinitely. Numbers formed by collecting ones, in the manner just explained, are called integers; they are also called integral, or whole numbers. 5. Numbers are divided into two classes, abstract and denominate. An Abstract Number is a number whose unit is not named; as, five, seven, eleven. A Denominate Number is a number whose unit is named; as, three pounds, six miles, seven months. Denominate numbers are sometimes called concrete numbers. A denominate number may be either simple or compound. It is a Simple Number when all the units of the collection are of the same name or denomination; as, eight yards, eleven ounces, five feet. It is a Compound Number when all the units of the collection are not of the same denomination; as, three feet and six inches, four hours and twenty minutes, two pounds and eleven ounces. NOTE. All integers are simple numbers. |