PREFACE. THE following work is the First Part of a treatise on Arithmetic, the plan of which is in many respects entirely new. The book commences with Mental Questions, designed to illustrate all the fundamental principles of the science, and of such a character as to be understood by children at a very early age. The addition, subtraction, multiplication and division, of units and tens, the reason for carrying the tens in each operation, the principles of simple and compound, proper and improper fractions, and the various operations that may be performed upon them, are all taught in the most simple and concise manner. After the pupil has become accustomed to exercise his judgment in the solution of these questions, he is allowed to read and write numbers embracing many denominations, and to perform examples on the slate. The assistance of the teacher is then required, to explain clearly and fully on the board, each new principle that is introduced, and with little, further aid, the pupil will be able to go through the remaining chapters. The boy who can read and write integers as high as millions, may be taught, with equal facility, to read and write decimals as low as millionths; and if Numeration be properly inculcated at first, no more difficulty will be found in the operations on decimals, than in those on whole numbers. The two are therefore very properly combined, -the necessity of a separate and unintelligible series of rules, is obviated, and the scholar learns readily to solve all questions that may be proposed to him. No rules are given to be learned by rote, but the parts that are of most importance are printed in italics. The teacher should assure himself, by frequent questioning, that every principle is fully understood, -the pupil being required to tell not only how, but why, every thing is done. The remarks given under the head of Examples for the Board, are designed as a guide to the teacher in his explanations to the class. If the scholar be left entirely to himself, he will not understand any assistance the book may give, nearly so well as if it were communicated verbally by his instructor. The difference between abstract and concrete numbers will be early perceived by the pupil, and it will be well for the teacher to 1* (5) • examine him on that subject, after he has mastered the first few sections. If similar examinations are continued throughout the whole course of the work, the scholar who has properly studied merely the present volume, will be able to solve all questions that occur in ordinary business, readily and understandingly. HINTS TO THE TEACHER. The object of the text-book should be merely, to serve as a guide, to point out the way, and assist the pupil, in removing any obstacles that may obstruct his progress. It is not sufficient, that the reader should have a tolerable understanding of its contents, or that he should be able to repeat by rote all its arbitrary rules; but he should be taught to form his own rules, and to apply the principles which he has himself discovered, in every possible manner. To give this full understanding, the aid of the teacher will be frequently required, and he will be enabled to give his assistance to the best advantage, by adopting a plan similar to the following, from which the author has derived great benefit. At the end of every lesson, both in the Mental and the Written Exercises, the books are all closed, and original questions are proposed by the teacher, to the class, and by each pupil to his lefthand neighbour. In this manner, the familiarity of each one with the subject of the lesson, is thoroughly tested, and the teacher perceives at once what farther explanation is required. The class are examined each day, on the preceding lesson, and are required frequently to review the part they have gone over. They are never allowed to enter on a new section, until they are perfectly familiar with the one they have left. If the pupil finds any question contained in the book, too difficult, he is required to work out the statement with smaller numbers, until he perceives the course to be pursued. At the recitations in Written Arithmetic, each scholar hands his slate to his neighbour. The proper mode of performing the sums is then stated, the correct answers are read from the key, and the errors marked on each slate, to be afterwards corrected by the owner. After the slates have been thus examined, one or more original questions are written on the board, to be solved and explained aloud, by different members of the class. PHILADELPHIA, 1844. THE ELEMENTS OF ARITHMETIC. MENTAL ARITHMETIC. [THE teacher should illustrate this and the succeeding sections by grains of corn, beans, or some other small articles. Let him make piles of ten, and show that two piles will be called twenty; three piles, thirty; ten piles, one hundred, and so on. In this way, the pupil will soon learn to count a thousand, without difficulty.] I.-1. Charles has one apple, and James gives him one more. How many does he then have? How many are one and one? 2. I have two pins, and my sister gives me one more. How many do I then have? How many are two and one? 3. George has three cents, and his father gives him How many has he then? Three and one are how many? one more. 4. How many fingers have you on one hand? How many thumbs? How many of both? Four and one are how many? 5. William has five plums in one hand and one in the other. How many has he in both hands? Five and one are how many? 6. Jane has six peaches, and Eliza has one. How many have they both? Six and one are how many? 7. If I ride seven miles and then walk one mile, how many miles shall I have gone? Seven and one are how many? 8. Eight boys are playing, and one more comes to join them. How many are there then? Eight and one are how many? 9. If I spend nine cents for fruit, and have one cent left, how many cents had I at first? Nine and one are how many? One, two, three, four, five, &c., are called numbers. Joining numbers together as we have done in the above examples, is called ADDITION, and by continuing to add, we may form numbers to any extent we please. There are various ways of expressing numbers: by words, as above; by letters, as was the custom among the Romans; or by the ten Arabic figures, One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Naught. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. 10. Write on the board the figures from one to ten. Count ten. The number ten is written by placing 1 at the left hand of 0, thus, 10. Twenty, or 2 tens, is written 20; Thirty, or 3 tens, 30; and so on to one hundred, or 10 tens, which is written 100. 11. The number one is called a Unit. How many units are there in 2? How many in 3? In 5? In 8? In 9? In 10? II.-1. I have 10 fingers and thumbs. If I had 1 more, how many should I have? How many are 10 and 1? 2. In a certain school 10 boys were studying Latin, |