PREFACE. In this little book I have endeavoured to expound, upon simple and rational principles, the rudiments of the SCIENCE OF ARITHMETIC. With rules I have given reasons: and although the work is designed chiefly for schoolboys and young persons, yet, contrary to the usual practice, I have chosen to regard the learner less as an arithmetical machine than as an intellectual being. I venture to hope, that what I have here done may meet with some degree of countenance from Schoolmasters and Teachers; and that it may also prove acceptable to the solitary and self-dependent student. This is not an unreasonable hope: for, although so-called Treatises on Arithmetic are very numerous, the number of books really deserving of the appellation is but few. As I have reserved no room here for even the most summary analysis of the following pages, I must leave them to the candid examination of those who may be interested in the progress of this class of educational books. I trust no fault will be found with me for the familiar and colloquial form of exposition I have adopted: any attempt at elevation of style, in works of this kind, is wholly misplaced. I have imagined my own pupils before me; and I have addressed them as I was formerly in the habit of doing in oral instruction. My sole aim has been to be intelligible, and to invest the subject with what interest I could consistently with the preservation of scientific accuracy: but if there be one part more than another, to which I would invite special attention, it is the subject of Decimals, towards the end of the book. LONDON, 1853. J. R. YOUNG. *A Key to the work is now published: besides solutions in full to all the Exercises, it will furnish some additional instructions for the otherwise unassisted learner. ARITHMETIC. (1.) THE marks used in Arithmetic are 1, 2, 3, 4, 5, 6, 7, 8, 9; which stand for one, two, three, four, five, six, seven, eight, nine.* These marks are called figures; and by help of these figures, and another mark, 0, to stand for nought, or nothing, any number may be written down. The mark 0 or nought, may also be called a figure; so that there are ten different marks or figures used in Arithmetic. (2.) You must remember that a figure is only one of these marks: when you see two or three or more of them written side by side, you see a number: thus, 24 is a number, of two figures; it stands for twenty-four; also 37 is a number of two figures; it stands for thirty-seven: and so on. Twenty-four means two tens and four; thirty-seven means three tens and seven. And in like manner forty-eight means four tens and eight; and this number, written in figures, is 48. You see then, that, in a number of two figures, the first, or left-hand figure, tells us how many tens there are in the number; and the second, or right-hand figure, tells us how many ones, or units there are in the number, besides the tens: one, you are to remember, is also called unit, or unity. (3.) From what has now been said, you see that the word number does not mean the same thing as the word figure: there are only ten different figures, or single marks, but by joining two or more of these together, we may write down as many numbers as we please. The single figures themselves are also called numbers, as well as sets of two, three, or more figures: thus, 5, 7, 6, &c., are numbers of one figure each; 57, 75, 76, &c., are numbers of two figures; and 576, 756, &c., are numbers of three figures. The number 57 is fiftyseven, the number 75 is seventy-five, the number 76 is seventy * Arithmetic may be defined as the science which teaches how to perform computations by numbers. It would perhaps be of but little use to a beginner to give a formal definition of Arithmetic in the text. B six, and so on. The number 576 is five hundred and seventysix, the number 756 is seven hundred and fifty six; and in any number of three figures, the first figure on the left hand tells us how many hundreds there are in the number; the next figure tells us how many tens there are, besides the hundreds; and the last figure tells us how many units there are, besides the hundreds and tens: there may be no units after the hundreds and tens; if so, a nought or 0, is put for the last figure: thus, five hundred and seventy, would be written in this way, 570; and seven hundred and fifty, Iwould be written 750. Should there be no tens after the hundreds, then, in the same way, a O is put in the place of tens: thus, 506, means five hundred, no tens, and six units; that is, five hundred and six; also, 605 means six hundred and five; and 600 means six hundred, without any tens or units besides. You see, therefore, that if you write down a single figure, you mean so many units; but if you put a nought to the right of it, you mean ten times as many, and if two noughts, one hundred times as many. (4.) When a number has four figures, as, for instance, the number 3562, the first figure on the left hand tells us how many thousands there are in the number; the next figure, how many hundreds besides; the third figure, how many tens; and the fourth, or last figure, how many units besides; so that the number just written is three thousand five hundred and sixty-two: if, instead of five hundreds, there had been no hundreds, the number would have been written 3062; that is, three thousand and sixty-two. In like manner, 3502 is three thousand five hundred and two; also, 3002, is three thousand and two; and 3000, is three thousand only. (5.) From the explanations which have now been given, you see that when a figure stands by itself, that is, without any figures beside it, it stands for a certain number of units: thus, 6 stands for six units, or six ones; and that when a figure does not stand by itself, but at the right-hand end of a row of figures, it still stands for units: thus, the 6 in the number 346 is still six units; but the 6 in the number 562 stands for ten times 6, or sixty; the 6 in the number 4637 stands for ten times sixty, or six hundred; the 6 in the number 6253 stands for ten times six hundred, or six thousand. It is on this account that we say that the last figure of any number occupies the place of units; the next figure, |