No. 1, contains only the five fundamental rules of Arithme tick. These rules have been treated of more largely than is customary, from the belief that most pupils pass from these to the more difficult rules before they are thoroughly acquainted with them. Fractions, and the Compound Rules, are entirely omitted in No. 1, until the learner is well acquainted with the working of whole numbers. No. 1, is also made small that the young learner may not be disheartened by having a large volume put into his hands; and that the parent shall not be under the necessity of purchasing a larger and more expensive book for his child before he shall require it. No. 2, commences with the Compound Rules, and includes all that is necessary of every other rule in Arithmetick for practical purposes, and the transactions of business. In No. 2, the EXPLANATIONS of the nature and principles of each rule are also fully and minutely given. No. 2, likewise, contains a Practical System of Book-Keeping. Tradesmen, without number, the most industrious and meritorious of men, often carry on their business with great difficulty, and many of them become involved and ruined, merely from the want of a simple system of keeping their accounts. Such a system is, therefore, given at the close of No. 2. One very important advantage of this work is, that all, or nearly all, the questions for practical exercise are in dollars and cents. The Author of the following work appeals, without apprehension or reluctance, to that publick, whose candour and liberality he has often experienced, to decide upon this attempt to render the elementary rules of Arithmetick both practical and popular, and also beneficial to the youth of this country. LYMAN COBB. New York, Jan. 25, 1832. COBB'S EXPLANATORY ARITHMETICK. Question. What is ARITHMETICK? Answer. Arithmetick is, in its practice, the art of reckoning, or of calculating numbers, and is comprised in the five following rules: Numeration, Addition, Substraction, Multiplication, and Division. Note.-To TEACHERS. All the EXPLANATIONS should be thoroughly and carefully read by the young scholar, as they are intended to impress deeply on his mind the principles of the rules, and their importance in his operations. EXPLANATIONS. Arithmetick is an art which is constantly in use in the transactions of trade; and, besides being employed in numerous operations of manufactures and of science, is a principal means by which the mariner regulates his course over the ocean. It is of almost universal use in the affairs of civil society, and a dexterous and correct practice of it is exceedingly valuable to the man of business, to the student, and to the gentleman. This art, however, valuable as it is, is scarcely more useful than it is pleasing to its possessor, in the power which it gives of calculating with accuracy and despatch. This desirable art is very easy of attainment, when the principles, and the practice of them, are properly placed before the attentive learner. Before the scholar enters on the study of Arithmetick, which, as I have said, is the art of reckoning, or of calculating numbers, he must devote a little attention to the mode of writing down and of reading these numbers, as they are the materials with which he has to perform his work. NOTATION AND NUMERATION. Q. What is NOTATION? A. Notation is the writing down, or noting the numbers used in Arithmetick, EXPLANATIONS. Notation and Numeration are very simple matters, quickly and easily learned; they are necessary as a first step in Arithmetick, and a clear and correct knowledge of them will greatly facilitate the learner's progress. I will state here, however, that I shall not trouble the learner with more of this branch of the subject than is necessary to the step he is taking. Notation and Numeration include the writing and the reading of all sorts of numbers; that is, of whole numbers, both simple and compound; of fractions, both vulgar and decimal; but, as in the commencement of this study, we have to deal with simple whole num bers only, I shall confine my remarks to these, reserving what is to be said of other numbers until I come to treat of the working of them. This is the course which I intend to pursue throughout this work, in which I shall carefully abstain from perplexing the learner with unnecessary matter, and shall endeavour to lead him on in the easiest and most pleasing manner to a knowledge of the uses and the power of figures, so far as they are required for the ordinary business of life, or as a preparation for higher mathematical and other studies. Q. How is Notation or writing numbers in figures performed? A. For the purpose of Notation, there are ten several forms or figures in use; and, with these ten figures only, any number, however large, can be clearly and conveniently expressed. Q. What are the names of these characters or figures? A. The names of the ten figures are as follow; and the value, or amount, which each of them serves to express, is written underneath in words, thus: 1 2 3 4 5 6 8 9 0 one, two, three, four, five, six, seven, eight, nine, naught, or cipher. EXPLANATIONS. The first nine figures are called significant figures, to distinguish them from the cipher, which, of itself, has no value, or it expresses nothing in amount; but, as it is placed, it may serve to increase or decrease the value of the figure or figures with which it is connected. The cipher is useful, however, and cannot be dispensed with when we write down the higher numbers, as will appear hereafter. It must be remembered that each of the other nine figures, when standing alone, unconnected with any of the others, or, when standing the last on the right hand of a series of figures, expresses merely the number which I have written underneath it; and the highest of these figures, in this its single and separate state, expresses a number no higher than nine; and yet, hundreds, thousands, and even higher numbers, are continually required to be expressed. The manner in which this important purpose is accomplished, is a changing of the place or places of these figures; by which changing, the value, or the amount, severally expressed by them, becomes changed. This change of place, however, of which I speak, is not a mere removal of the figure, or figures; for a mere removal, as has been stated before, would not alter the value of any of them. But the change in the value is produced by ranging two or more of the figures together, so that they stand in a kind of rank, one before another, and it is according to the RANK or STATION which it occupies that the value of any, and of every figure is estimated. Note.-To TEACHERS. In order to fix in the mind of the young scholar the effect of this changing the situation of figures, let him write down, neatly and clearly, on a card, the ten figures: then cut the card into square pieces, each containing one of the figures. Having the figures so prepared, let the scholar take one of them, the first, for instance, and place it before him on the table; thus standing, single, alone, the figure expresses the number, one. Then let him take another of the figures, (let it be the second figure,) and place it at the |