PREFACE. IN constructing an Arithmetic for the use of schools, two errors are to be avoided; the total exclusion of explanation, and the redundancy of it-the laying down of naked arbitrary rules, as if the scholar were incapable of comprehending the reason of them; and that minuteness and excess of explanation, which leaves nothing for his own discovery and investigation. Explanations should, in the judgment of the Author, be applied to the principle of the rules, rather than to the solution of particular questions; and nothing should be done for the learner, which he is capable of doing for himself. Every one recollects the satisfaction, with which, as he advanced in his mathematical course, he found himself able to take the progressive steps by himse f, without the aid of his teacher or his fellowstudents; and every one is conscious of the surer grasp, he yet retains, on the acquisitions he made by the unassisted exercise of his own faculties. To the mind as to the limbs, leading strings may be of service in taking the first steps; but, if never laid aside, there can never be confidence in the ability to go alone. In the preparation of this treatise, the Author has aimed especially at brevity, clearness and precision; and he has added only so much of theoretical explana tion, as will enable the scholar to understand the principle and reason of the several rules, and methods of operation. A new method of statement in Proportion, and several new and convenient contractions, under different rules, will be found in this book. The reader is referred to Decimals, to the theoretical explanations of Proportion, the Square and Cube Roots, and to Mensuration, as containing original matter, not to be met with in other systems. A brief practical system of Book-keeping is subjoined, which will be found sufficient for all the purposes of ordinary business; and indeed, for mercantile establishments, those only excepted, whose extensive operations require the use of the more complicated system of double entry. As the best preparation for the use of this book, or of any other treatise of the same grade, the Author would recommend a thorough acquaintance with CorBURN'S MENTAL ARITHMETIC. The effect of this invaluable 'ttle work, in disciplining the minds of younger pupils to habits of clear and accurate thinking, and the aptitude and rapidity it gives in numerical calculations, are known to all who have used the book. A word in conclusion, on the method of teaching Arithmetic. No school should be without a blackboard The scholars should be formed into classes, every member of which, should in rotation, be required to work a question on the black-board, in the presence of his class, and to explain each step of the process; and the reason of it. If taken through the book in this way, it is hardly possible, that a pupil should fail of attaining a competent knowledge of the science. ARITHMETIC. ARITHMETIC treats of the properties, relations, and combination of numbers. Its principal branches are, Notation or Numeration, Addition, Subtraction, Multiplication, and Division. Numeration teaches to write and read numbers. METHOD OF NOTATION. The characters by which numbers are expressed, were derived from Arabia. They are the following: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. By variously combining these, all possible numbers are expressed. The simple characters, however, carry us no higher in numbering than nine units. To denote an additional unit, the first figure is repeated, but in a higher place, having the cipher (which by itself has no value) on the right: the 1 then becomes a unit of the second order, and has ten times its simple value. The addition is continued up to nineteen, by placing the simple. figures at the right of this unit of the second order, in the room of the cipher: thus, 11, 12, 13, 14, 15, 16, 17, 18, 19; which may be read, one ten and one, one ten and two, &c. When all the simple characters are repeated in this combination once, the addition is made by increasing the unit of the second order one, (making it two tens,) and repeating the cipher: thus, 20. Again the simple characters are added successively in the same manner as before, and we have 21, 22, 23, &c. When the whole are repeated, the second order is again increased, and becomes 30. After the nine digits have all in succession been used in the second crder, we are carried in numbering up to 99, or, nine tens and nine; then the additional number is expressed, by removing the original unit one place farther to the left, where it becomes a unit of the third order, and has ten times the value which it had in the second, and one hundred times its simple value: it is then named hundreds 5 and all the preceding combinations are repeated successively with it, up to 199, when this unit of the third order is increased, and becomes 200. The simple figures being used in succession, as units of the third order, bring us to 999, when another remove toward the left, gives them a tenfold higher value, and makes them thousands, (1000.) All the preceding combinations arc then repeated, with their own appropriate names, and that, of thousand superadded: the notation thus carries us to six figures, ending with 999.000-(that is, nine hundred and ninety-nine thousand.) Then succeed three more places of figures, read like the first three, with the name of million superadded. Thus the name changes with every additional three places toward the left. Three places of figures commencing at the right form a period, to which a distinctive name is given, as follows: Units, Thousands, Millions, Billions, Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions. Every period is read precisely alike, except that a different name is added at the end of each. I. To read numbers, therefore: RULE. Beginning at the right, divide them into periods of three figures, and then read from the left each period by itself, adding to each its appropriate name. EXAMPLE. 80.652.941.600.807.362.546.278.009.650.208 Which is read, eighty nonillions, six hundred fifty-two octillions, and so on to the last period, to which the name (units) is not added. Divide off by points, and read the following numbers: 5461 412316 4666240 987000000 60059821102700836004 II. To write numbers. RULE. Beginning at the left hand, write by periods, placing each period in its proper order; taking care to supply by ciphers, those periods and places, that are omitted. in the question. If, for example, the number to be written, be ninety billions, four hundred and sixty-one thousand and twelveyou begin at the left hand, and write the billions (90.) Next to billions comes the period of millions; but as there are no millions named in the question, you fill up that period with ciphers. Thousands follow millions, which you write (461), and units close the series. But as there are only two places of units in twelve, you fill out that period by a cipher in the place of hundreds, (012.) The whole number stands thus: 90.000.461.012. Write the following numbers: Nine millions, seventy-two thousand, and two hundred. Eight hundred millions, forty-four thousand, and fiftyfive. Eight billions, sixty-five millions, three hundred and four thousand, and seven. Fifty-four sextillions, three hundred trillions, sixty-seven millions, four hundred and twenty. Seventy decillions, two hundred and thirty-one octillions one billion, one hundred thousand, and three hundred. From the system of notation already explained, it is evident that figures have a simple value and a local value. When a figure stands in the place of units, it has a simple value. When a figure does not stand in the place of units, it has a local value, which varies according to its distance from the unit's place. In the annexed series, the first right hand figure has a simple value, the second has a local value tenfold greater: the third a still higher local value, tenfold greater than the second; and so on, increasing in the same proportion with every additional remove toward the left. C Thousands X Thousands Thousands 1 1 1 1 1 Units 1 1 2 2 2 2 2 2 2 |
