ADVERTISEMENT TO THE SECOND EDITION. 34 1837 THE object of the following Treatise, is to present a full and regular course of whatever is useful in Arithmetic. With this view, rules are given for performing all the requisite operations, and are illustrated by many examples; and numerous exercises are prescribed, to afford the pupil that practice which alone can produce expertness and accuracy in the management of numbers. The Definitions and Rules, which it may perhaps be proper for the learner to commit to memory, are in the largest type employed in the work; the Examples and Exercises, and the principal illustrations, are in a character somewhat smaller; the less important illustrations and remarks are in a type still smaller, and may perhaps be omitted by the younger pupil; and the notes contain miscellaneous information, which may often be interesting to the reader. The reasons of the rules and operations are explained, not in strict, formal demonstrations, but generally by simple and easy illustrations of particular cases and examples; and it is hoped, that the subject will thus be rendered as intelligible and attractive as possible. This part of Arithmetic is too generally neglected, both in treatises on the subject, and in teaching: and thus one of the principal divisions of Mathematical Science is converted into a mere practical art; and what is calculated to call forth and improve the reasoning powers of the pupil, is degraded into a dry exercise of memory. Of the Examples and Exercises, some are proposed in purely abstract terms, being intended merely to afford practice to the learner in the rules and modes of calculation. To these are subjoined, in those parts of the work in which it could be conveniently done, other questions, which will not only afford the pupil farther exercise on the rules which precede them, but will also furnish him with many important facts in Commerce, Geography, Astronomy, Chronology, Chemistry, and other departments of knowledge. As the information contained in these questions has been all derived from authentic sources, its correctness may be depended on; and it is hoped, that what is thus presented may excite in the young reader a desire to enrich his mind, by the acquisition of farther information of a similar nature. Some subjects are omitted, which in several works on Arithmetic are treated at considerable length. Such is Barter,-an application of the Rule of Proportion, which is of scarcely any use as a part of Mercantile Arithmetic. Neither has the method of calculating Annuities at Simple Interest been introduced; as it is unjust in principle, and, in real transactions, all calculations respecting annuities are made according to the principles of Compound Interest. Many applications of the rules to the circumstances of bodies in motion, and to other subjects in Natural Philosophy, which have been given in works on this science, have also been omitted, as they require a degree of knowledge to render them intelligible and useful, which is to be acquired only by the study of a considerable course of mathematical and physical science. Instead of these, however, other subjects have been introduced, which are little known, but which cannot fail to be interesting to the more advanced pupil. Of this kind are Continued Fractions, and the Theory of Arithmetical Scales. The principal and most important parts of the work, also, bave been explained and exemplified more largely than the rest; and in these parts a greater number of exercises have been left unwrought, for the improvement of the pupil. Thus, the Simple and Compound Rules, Proportion, Practice, Interest, and Exchange, have M306030 been treated at considerable length; and much care has been taken to render them In preparing the present edition for the press, considerable changes have been As so much eare and labour have been employed in improving this edition, it A TREATISE ON ARITHMETIC, IN THEORY AND PRACTICE. NOTATION AND NUMERATION. ARITHMETIC is a science which explains the proper ties, and shows the uses of numbers. Unity, or a unit, is the number one. Every other number is an assemblage of units.* For facilitating the management of numbers in Arithmetic, they are generally expressed by signs or characters. The method of expressing numbers by characters, is termed Notation. The method of discovering the values of numbers already expressed by characters, is called Numeration. In modern arithmetic, all numbers are expressed by means of ten characters: the cipher, or zero, 0, which has no value; and the nine significant figures, or digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, which denote respectively the numbers one, two, three, four, five, six, seven, eight, nine. When any of these figures stands by itself, or when it is followed by no other figure, it expresses merely its simple value; but when it is followed by one figure, it signifies ten times its simple value; when by two, one hundred times; when by three, one thousand times; and so on, by a tenfold increase for each figure that follows it.The increased value thus denoted by a figure in consequence of its position, is called its local value. * This definition, though it might appear otherwise, is applicable to fractions as well as to whole numbers. Thus, the fraction is an assemblage of seven units, each of which is one-tenth of the integer. The use of the term unity to denote the number one, appears to be improper; if it be, however, it has the sanction of long custom. Some late writers, probably to avoid this use of the word, employ in the same sense the term unit, but without the indefinite article; perhaps because it is not used before two, three, ten, &c. But, besides the sanction of custom, may not the article be used before it with the same propriety as before the words hundred, thousand, million, &c.? It seems to be as awkward, and as incorrect, to talk of increasing a number by unit, as of increasing it by hundred or thousand. B Thus the figure 5, when it stands by itself, or is followed by no other figure, denotes simply five; but when placed to the left of one figure, it expresses ten times five, or fifty; when to the left of two figures, it expresses ten times fifty, or five hundred; when to the left of three, ten times five hundred, or five thousand, &c.; as in the number 5555. In like manner, in the number 7854, the 4 signifies simply four units, or four; the 5, five tens, or fifty; the 8, eight hundred; and the 7, seven thousand. The names of the local values of figures will be known from the following table : From this table it appears, that if a line of figures be divided into periods of three figures each, commencing at the right hand, the first period will contain units, the second thousands, the third millions, &c. ; and it is usual and convenient thus to divide the figures by which large numbers are expressed, for the purpose of facilitating their numeration The periods succeeding those contained in the table, are quintillions, extillions, septillions, octillions, and nonillions; and analogical names might be formed for the still higher periods. Those already given, however, are more than sufficient to express any number which it is ever necessary to designate in language. The local value of any figure used in expressing a number, is at once discovered from this table. Thus, 6 in the eighth place from the right hand, expresses six tens of millions, or sixty millions; and, conversely, sixty millions will be expressed by the figure 6 in the eighth place.* * This method of dividing lines of figures into periods, and of naming those periods, is that which is used by the French and Italians. It is strongly recommended by its simplicity and elegance; and has been adopted in the treatise on Arithmetic by Mr. Anderson, in the Edinburgh Encyclopedia; and in Professor Leslie's "Philosophy of Arithmetic." In other English works, the periods are made to consist of six figures each, and have the same names as those in the table given above, except thousands, for which there is not a distinct period. The two methods agree as far as hundreds of mil. lions, and it is rarely necessary to name larger numbers. For the use of those who pre The cipher, or zero, having no value, is used in combinations of figures, to fill places where no value is to be expressed, and thus to make the other figures occupy those places in which they will express the intended values. Thus the figures 365, combined in this order, denote three hundred and sixty-five; but the expression 306050, which contains the same significant figures, means three hundred thousand, no tens or thousands, six thousand, no hundreds, five tens, and no units; or three hundred and six thousand, and fifty.-From these principles we have the following rule : To express in Words the Numbers denoted by Lines of Figures. RULE (1.) Commencing at the right hand side, divide the given figures into periods of three figures each, till not more than three remain. (2.) Then the first period towards the right hand contains units or ones; the second, thousands; the third millions, &c. as in the Numeration Table: and therefore commencing at the left side, annex to the value expressed by the figures of each period, except that of the units, the name of the period. Thus, the expression 37053907, becomes by division into periods, 37,053,907, and is read thirty seven millions, fifty three thousand, nine fer the English method, the following table is subjoined; and the answers of the Exer. cises are given according to both methods at the end of the work. It is scarcely necessary to say, that the rules and directions given in the text will be applicable in this method, if the periods be made to consist of six figures each instead of three, and if the second period be called millions, the third billions, &c. as in the following table:- |