ARITHMETIC. ARITHMETIC is the science of numbers. It explains their properties, and teaches how to apply them to practical purposes. The principal or fundamental rules are, Notation, Numeration, Addition, Subtraction, Multiplication and Division. These are called fundamental rules, because all questions in Arithmetic are solved by one or more of them. Notation is the expressing of any number or quantity by figures; thus, 1 one; 2 two; 3 three; 4 four; 5 five; 6 six; 7 seven; 8 eight; 9 nine; 0 cipher. Figures are sometimes called digits; they were formerly all called ciphers; hence, the art of Arithmetic was called ciphering. There are two methods of Notation, the Arabic, as above, and the Roman, which is expressed by the following seven letters of the Alphabet. I, V, X, L, C, D, M. 1 2 3 4 5 6 7 8 9 10 20 30 40 I, II, III, IV, V, VI, VII, VIII, IX, X, XX, XXX, XL.. 50 60 70 80 90 100 500 1,000 L, LX, LXX, LXX, XC, C, D, M. When a letter of less, is placed before one of a greater value, it diminishes the value of the greater, by the value of itself-thus, X signifies ten, but IX is only nine. When a letter of less, is placed after one of greater value, it increases the value of the greater by the value of itself. This method is seldom used except in numbering chapters, sections, &c. QUESTIONS. 1. What is Arithmetic? 2. What are the principal, or fundamental rules? 3. Why so called? 4. What is Notation? 5. What are figures sometimes called? 6. What were they formerly called? 7. How many methods of Notation, and what are they? 8. How many are the Arabic characters, or figures? 9. By what is the Roman method expressed? 10. How is a letter affected, when one of less value is placed before it? 11. How when one of less value is placed after it? 12. For what is the Roman method of notation principally used? B NUMERATION. Numeration teaches to express in words the value of any number represented by figures. Thus, 365 is read, three hundred and sixty-five. Figures have a simple and relative value. When a figure stands alone, its value is simply so many units, or ones; as, 2 two; 3 three; 4 four. Their relative value is derived from the place they occupy when joined together, or from their distance from the unit's place. Thus, 2 and 3 express their own value; simply so many units; but they are made to express either 23 or 32; that is, either three units and two tens, or two units and three tens. Hence, it appears, that the first or right hand place always expresses so many units; it is therefore called the unit's place; the second, the place of tens, expressing always as many tens as the figure contains units. The third place is hundreds -the fourth thousands, as may be seen by the following TABLE. Tens Thousands Tens of thousands 3 2 1Three hundred and 21 ,4 3 2 1 Four thousand and 321 5,4 3 2 1 Fifty four thousand and 321 ,7 6 5,4 3 2 1 7 millions 654 thou. 321 8,7 6 5,4 3 2 1 87 millions 654 thou. 321 The cipher, when standing alone, or at the left hand of another figure, signifies nothing, as 05, 005, is five in either case, because it still occupies the unit's place. But when placed at the right hand of another figure, it increases its value in a tenfold ratio, by removing the figure farther QUESTIONS. 13. What is Numeration? 14. What is the value of a figure standing alone? 16. From what is their relative value derived? 17. What does the first, or right hand figure, always express, and what is it called? 18. What is the value of the cipher, when standing alone, or at the left hand of another figure? 19. What effect has it when placed at the right of another figure? from the unit's place. This may be seen by the following TABLE II. 0 Nothing. 20 Twenty. 200 Two hundred. 2000 Two thousand. 20,000 Twenty thousand. 200,000 Two hundred thousand. 2,000,000 Two millions. To know the value of any number of figures. RULE 1. Numerate from the right hand to the left, by saying units, tens, hundreds, &c., as in the Table. 2. To the simple value of each figure join the name its place, reading from the left hand to the right. HHMMMMM The first division of the foregoing Table is according to the French method, into periods of three figures each; the name of the period is superadded. The second division is according to the English method, into periods of six figures each; the name of each period is subjoined. The two divisions of the Table agree for the first nine figures beyond that they assume different names. principles of Notation in both are the same. In the former method, the names, units, tens, hundreds, are repeated in each period; in the latter method, thousands, tens of The QUESTIONS. 20. How may the value of any number be found? 21. What re the two methods of numeration in the third table? 22. In what respect do hey differ? thousands, hundreds of thousands, are repeated with the name of the period. Let the scholar point the following numbers into periods and read them. 3445 67891 983452 5437643 67821356 436543897 5678923412 96754329876 1234678901268 Express the following numbers in figures. 1. Twenty-three. 2. Thirty-five. 3. One hundred and twenty. 4. One hundred and twenty-six. 5. Ten thousand three hundred and twenty. 6. One million two hundred and twenty thousand, three hundred and forty-five. 7. Two billions, twenty seven millions, three hundred and forty thousand, four hundred and seventeen. 8. Forty five quadrillions, six hundred and twenty trillions, four hundred and twenty billions, four hundred and twenty millions, two hundred and twenty thousand, three hundred and nineteen. The scholar should be well versed in Notation and Numeration, before proceeding to the following questions. EXERCISES. 1. If John has 6 apples, and his brother gives him 3 more, how many will he have? 2. James being on a visit at his uncle's, one of his cousins gave him 3 walnuts, another 4, and his uncle gave him 9, how many did he receive, and how many more did his uncle give him than his cousins? QUESTION. 23. In what should the scholar be well versed before proceeding any further? 3. Samuel bought a book for 15 cents, and sold it for 17, how many cents did he gain? 4. If a boy pay 15 cents for a book, 10 for a knife, and 6 for a dozen apples, how many cents does he pay in all, and how many more for the book than for the knife, and how many more for the book and knife than the apples? 5. If an inkstand cost 10 cents, an orange 5, a lemon 3, and a dozen of quills 14 cents, what is the cost of the whole, and how much more will the inkstand, orange and lemon cost than the quills? 6. A man bought of a drover 3 sheep and a cow; for one of the sheep he paid 4 dollars, for the other two he paid 3 dollars apiece, for the cow he paid 20 dollars; how many dollars did he pay for the whole, and how much more did he pay for the cow than for the sheep? 7. Joseph bought a sled for 25 cents, a yoke for 12 cents, and a whip for 6 cents; what did they all cost him, and how much more did the sled cost than the yoke and whip? S. If I pay 6 dollars for a hat, 8 for a cap, 4 for a vest, and 14 for a coat, what do I pay for the whole, and how much more do the coat and vest cost, than the hat and cap? 9. If I owe one man 6 dollars, another 8, another 12, another 20, how much do I owe in all? 10. The scholars in a certain school are divided into 4 classes in the first class there are 10 scholars, in the second 12, in the third nine, and in the fourth 14, how many in all ? 11. If from my library, I lend to one man 5 books, to another 10, to another 8, to another 12, to another 20, how many do I lend in all ? 12. In my garden there are 6 apple-trees, 8 pear-trees, 10 peach-trees, 18 plum-trees, how many trees are there in all? 13. In a certain school, 10 study music, 12 French, 14 Spanish, how many are there, and how many more study French and Spanish, than music? 14. Eliza had 4 finger-rings, Mary had 10, and Susan had 7: how many more had Mary than Eliza, and Susan than Mary, and how many had they in all? 15. A certain man had 4 boarders; for two he received |