the easy understanding of the pupil, while their solution requires only such arithmetical operations as he has already learned. Attention is respectfully called 1. To the simple treatment of the decimal system of notation, and the great number of exercises intended to familiarize the pupil with the facilities for calculation which it affords. 2. To the multiplicity of short exercises that can be performed without the aid of the pencil, or that require but few figures in their solution. Longer ones are not wanting to test the perseverance of the pupil. 3. To the directions to the pupil, having in view the formation of right habits of computation. The "making up " method of subtraction, and the so-called "continental" method in division, though not obtrusively presented, are worthy of the attention of teachers. The latter furnishes an excellent mental exercise. 4. To the suggestions for original problems, now so commonly resorted to by the best teachers to stimulate the interest of their pupils, and to give them a better understanding of the subjects to which they relate. Their usefulness as short practical exercises in penmanship, spelling, and composition, will be appreciated by all. 5. To the simple and direct methods of treating the fundamental rules, common and decimal fractions, percentage, interest, proportion, square and cube roots, the problems of mensuration, etc. 6. To the rigorous adherence throughout the work to the inductive methods of instruction. The number and variety of exercises and problems in this work are so great as to supersede any necessity for a supplementary book of exercises. It is earnestly recommended that the pictured illustrations may be regarded as merely suggestive of the objective demonstrations which the student should be encouraged to get up for himself. As far as possible, let the learner furnish all the apparatus needed. While he is engaged in preparing it, the principles to be illustrated will present themselves to his mind more forcibly than in the repetition of definitions and rules in which he can take but slight interest till he appreciates their significance. If this course be taken, the pupil will, in most cases, be able to make out his own analyses. These may be crude at first, but they will be the better for being his own. Observation and experience will guide to better forms. Such a method will give him a mastery of the subject, develop mental power, and cultivate a taste for independent investigation. NEW YORK CITY, May 15, 1886. 1. The signs 1, 2, 3, 4, 5, 6, 7, 8, 9, are called the nine digits; because first used to represent a number of fingers. The word digit is sometimes used for the word finger. The following are the written forms of the digits: 1. 2. 3. 4. 5. 6. Tens and Units. 7. 8. 9. 2. When we count more than nine we begin to count by tens and ones. After nine we say ten, then eleven, which means ten and one, twelve (ten and two), thirteen (ten and three), fourteen (ten and four), etc., to nineteen (ten and nine). Then we come to twenty, which means two tens, twenty-one (two tens and one), etc., after twenty-nine we have thirty, forty, fifty, etc. Counting. Count the balls of the numeral frame or other objects from ten to ninety-nine, as follows. ten and one ten and two ten and three etc., to ten and nine two tens two tens and one two tens and nine three tens and one three tens and two three tens and three etc., to three tens and nine four tens, etc. Writing. We may write these numbers by using the digits 1, 2, 3, etc., instead of the words one, two, three; thus: 3. But the writing of numbers is still further shortened by omitting the words ten and, or tens and; thus, for 1 ten and 1 we write 11; for 1 ten and 2 we write 12; for 2 tens and 1 we write 21, and so on up to 99 (9 tens and 9). Thus we can tell whether a digit stands for ones or tens by the place it occupies. If it represents ones, it has the first place at the right; if tens, it occupies the next on the left. |