The system of Numeration introduced will be found somewhat peculiar. In the first place, numbers are resolved into the natural periods of units, tens, hundreds, thousands, &c.; then, by combining periods of units and tens, all numbers between 1 and 99, inclusive, are formed; also, by combining periods of units, tens, and hundreds, all numbers between 1 and 999, inclusive, are formed; and so on through the higher denominations. The pupil is thus instructed not merely in the art of reading figures, but of forming and analyzing numbers. In Simple Addition, a large amount of practical work is presented for the scholar in his first efforts upon the slate. A peculiarity in this part of the system consists in adding by combination; by which mental labor is much dimin ished, while increased expedition and accuracy are secured. This is a matter of practical importance, both in the schoolroom, and the business operations of every-day life. A second peculiarity will be found in the application of the principle of Subtraction. After the introductory mental and slate exercises, a business application of the principle is introduced by natural and obvious gradations. The first consists in subtracting a single number from the amount of several numbers; the second, in subtracting the amount of several numbers from a single number; and the third, in subtracting the sum of several numbers from the amount of several others. These three distinct subdivisions, embrace every possible application of the principle of subtraction, and for each, specific directions, illustrations, and explanations are given. The same general system is carried out in the application of the other simple rules. The simple rules are succeeded by the decimal system, which is regarded as an extension of the system of integral numbers, to numbers of less than integral value, and susceptible of being used in immediate connection with them. This section is followed by Federal numbers, and this, in turn, by Fractional numbers, of each of which a practical application is made. The fundamental principles of numbers are thus introduced and applied to practical purposes in every possible form. This treatise, although designed especially for the pupil in his first efforts on the slate, is believed to embrace an amount of arithmetical matter, sufficient for the practical purposes of life. It is also strictly analytical. In conclusion, we add a single remark. Judging from the character of many books designed for schools, especially of such as are of a more primary character, the conclusion is natural, that their authors regard it as an object of the first importance, to adapt the language employed, strictly to the present capacities of the child; that is, to adopt the child's vocabulary as the language of their books. In this particular, we beg leave most respectfully to differ. The child is not always to remain such. The law of his nature is progress,-a gradual development of intellectual energy. Hence he needs something above his present capacity; something that shall elicit his intellectual energies. His course in life is not to be a railroad level ;—nor is intellectual speed the chief object to be attained. There must be substantial growth, and especially in the acquisition. and use of language. In the character of the language adopted in the preparation of the present treatise, we have had in view this acquisition; and, while we have written for the child, we have preferred to employ language adapted to the gradual development of youthful powers; and should the scholar find it necessary to consult his teacher or his dictionary occasionally, to learn the true import of a word, we apprehend the time so employed will not be valueless in its result. C. TRACY. NEW YORK, 1850. To subtract one Number from the sum of two, three, or more Numbers To subtract the Sum of two or more Numbers from a Single Number. To subtract the Amount of Several Numbers from the Amount of Several ARITHMETIC. § 1. Arithmetic teaches the nature and use of figures. 1. Figures are characters employed to represent numbers. 2. Operations by numbers, when performed in the mind unaided by the pen or pencil, constitute mental arithmetic. 3. When, however, these operations are expressed by written characters, the work is termed written arithmetic. 4. Both mental and written arithmetic embrace the same operations; viz. Numeration, Addition, Subtraction, Multiplication, and Division. NUMERATION. § 2. Numeration teaches the art of reading figures. SERIES FIRST, Composed of units. 1. A unit consists of a single object, not limited in its kind. 2. Hence, unit figures are the representatives of in 1. What does arithmetic teach? What are figures? What, mental arithmetic? What, written arithmetic ? What do mental and written arithmetic both embrace? 2. What does numeration teach? A unit consists of what? Unit figures represent what? dividual objects. They are presented in the following table : TABLE OF UNITS. 1. One unit is counted, one; and written 2. Two units are counted, one, two; and written 3. Three units are counted, one, two, three; and written 6. Six units are counted, one, two, three, four, five, six; 7. Seven units are counted, one, two, three, four, five, seven; and written and 6. six, 7. 8. 8. Eight units are counted, one, two, three, four, five, six, seven, eight; and written 9. Nine units are counted, one, two, three, four, five, six, seven, eight, nine; and written 9. 3. The above is called the table of units, because each figure represents single objects. 4. The single objects represented by each figure are determined by counting that number. § 3. How many whole objects are represented by figure 1? by figure 2? figure 3? figure 4? figure 5? figure 6? figure 7? figure 8? figure 9? What character is employed besides the nine given in the above table? Ans. The cipher. What is the form of the cipher? It resembles the letter O. Has it any value when standing alone? Ans. None. How can you determine the number of whole objects expressed by each figure? Ans. By counting the number represented by that figure. Write on the slate or board 3, 5, 7, 9, 2, 4, 6, 8, 1. One, how written ?-counted? Two, how written?-counted? Three, how written?-counted? Four, how written ?-counted? Five, how written ?-counted? Six, how written?-counted? Seven, how written counted? Eight, how written?-counted? Nine, how written?-counted? Why called table of units? How deter mine the objects represented by each figure? |