into the habit of depending upon them. A Key for the teacher's use will prevent any inconvenience at recitation. A "Practical" Arithmetic should deserve its name, and we have kept this in view throughout. We have asked, What applications of Arithmetic is the pupil likely to need in life? What are the shortest methods, and those actually used by business men? The branches of Mercantile Arithmetic have received special attention,-the making out of bills, the casting of interest, partial payments, operations in profit and loss, averaging accounts, equation of payments, &c. Much collateral information on business subjects has been embodied. In a word, the author has weighed every line, with the view of giving what would be most useful and best prepare the learner for the duties of the counting room. The great distinguishing feature of this book is that it is adapted to the present state of things. The last five years have been five years of financial changes; specie payments have been suspended, prices have doubled, the tariff has been altered, a national tax levied, &c. No Arithmetic that ignores these changes should be placed in the hands of our youth. Time is too precious to be wasted in learning things wrong, only to unlearn them on entering into active life. Our examples are adapted to the present: the prices given are those of to-day; the difference between gold and currency is recognized and taught; the rates of duties agree with the present tariff; the mode of computing the national income tax is explained; a full description is given of the different classes of United States securities, with examples to show the comparative results of investments in them. These are matters that children, as well as adults, ought to know and understand. It is hoped that these, with other features that will be obvious on examination but need not be mentioned here, may commend the work to teachers generally. NEW YORK, August 10, 1866. PRACTICAL ARITHMETIC. CHAPTER I. NUMBERS. 1. One, a single thing, is called a Unit. 2. If we join another unit to ONE, we have two; if another, THREE; and so, adding a unit each time, we get FOUR, FIVE, SIX, SEVEN, EIGHT, NINE. 3. One, two, three, &c., are called Numbers. A Number is, therefore, one unit or more. 4. Arithmetic treats of Numbers. 5. Numbers are either Abstract or Concrete. They are Abstract, when not applied to any particular thing; as, one, eight. They are Concrete, when applied to particu lar things; as, one pound, eight dollars. 6. That to which a concrete number is applied, is called its Denomination. In the last example, dollars is the 7. Counting is naming the numbers in order; as, one, 8. We may express numbers by writing out their names, as one, two, three; or by characters, as 1, 2, 3. QUESTIONS.-1. What is a single thing called?-2. What do we get by successive additions of a unit to one?-3. What are one, two, three, &c., called? What is a Number? 4. Of what does Arithmetic treat?-5. How are numbers distinguished? When are they called Abstract? When, Concrete ?-6. What is meant by the De- nomination of a concrete number?-7. What is Counting ?-8. How may we express acters. 11. The Arabic Notation is so called because it was introduced into Europe by the Arabs, who obtained it 1 2 3 4 5 6 g 8 9 NAUGHT ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE 12. The first of these figures, 0, is called Naught, Cipher, or Zero. It implies the absence of number. The other nine are called Significant Figures, or Digits, each signifying a certain number. 13. The greatest number that can be expressed with one figure is nine, 9. For numbers above nine, we com- First, 1 is placed at the left of each of the ten figures, forming 10, ten; 11, eleven; 12, twelve; 13, thirteen; 14, fourteen, 15, fifteen, 16, six. teen; 17, seventeen; 18, eighteen; 19, nineteen. Then 2, forming 20, twenty; 21, twenty-one; 22, twenty-two; 23, twenty-three; 24, twenty-four; 25, twenty-five; 26, twenty-six; 27, twenty-seven; 28, twenty-eight; 29, twenty-nine. Then 3, forming 30 (thirty), 31, 32, 33, 34, 35, 36, 37, 38, 39. Then 4, forming 40 (forty), 41, &c. Then 5: 50 (fifty), 51, &c. Then 6: 60 (sixty), 61, &c. Then 7: 70 (seventy), 71, &c. Then 8: 9. What is Notation ?-10. How many systems of notation are used? What are |