a rational and perspicuous unfolding of the little intricacies of the Art. And, amongst those designed for the instruction of the young, besides every defect of style, I have marked an entire absence of all that is worthy of the name of principle: a total want of what may properly be called, the REASON of the thing. And, whilst there is in them, nothing to satisfy and to inform the understanding of the learner, there is every thing to puzzle and discourage him.

Similar to these are the defects of every work that I have found on the subject of Book-keeping. It is not that the authors of these works do not themselves understand the subjects; but, it is simply because they have not the faculty of communicating, with the pen, the knowledge they possess.

Tradesmen without number, the most industrious and meritorious of men, carry on their business with great difficulty, become involved and ruined, merely from the want of a simple system of keeping their accounts.

To furnish them with such a system, and to communicate to them a knowledge thereof, in a very small Pamphlet, shall be my next endeavour.

Liverpool,

Νου. 10, 1829.

THOMAS SMITH.

For the purpose of easy reference, the Paragraphs, and not the Pages, are numbered.

INTRODUCTION.

1. Arithmetic, in its practice, is the art of reckoning, or, of calculating numbers. An art which is constantly in use in the transactions of trade, whether in rude or in civilized states of society; and, besides being employed in numerous operations of manufactures and of science, it is a principal means by which the mariner regulates his course over the ocean. This art, indeed, is of almost universal use in the affairs of civil society, and a correct, and a dexterous practice of it, must always be extremely valuable to the man of business, to the gentleman, and to the student.

2. Such is the value, such the importance of the art I propose to treat of; an art which, 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. Nor will it be foreign to my purpose to state, that this desirable art is easy of attainment; that is to say, easy, when the principles and the practice thereof are properly placed before the attentive learner.

3. Arithmetic, I have said, is the art of reckoning, or, of calculating numbers. Before we enter on the study of it, therefore, we must devote a little att

tion to the mode of writing down, and of reading these numbers, being, as they are, the materials with which we have to perform our work.

4. This writing down, or note-ing, the numbers, used in Arithmetic, is called NOTATION; and the reciting, or reading of them, in words, is called Numeration. These are very simple matters, quickly learned; they are necessary as a first step in our proceedings; and a clear knowledge thereof will greatly facilitate the learner's progress.

5. Here, however, let me state, that, at this stage of our proceeding, I shall trouble the learner with no more of this branch of our subject than is necessary to the step we are taking. Numeration and Notation include the reading and the writing of all sorts of numbers; that is to say, of whole numbers, both simple and compound, of fractions, both vulgar and decimal; but, as, in the commencement of our study, we have to deal with simple whole numbers only, so it is to these that I shall now confine myself, reserving what is to be said of other numbers, until we come to treat of the working of them. And this is the course intended to be pursued throughout this work, in which I shall carefully abstain from perplexing the learner with unnecessary matter, and shall study to lead him on in the easiest and most pleasing manner, to a knowledge of the uses and the powers of figures, so far as they are required for the ordinary business of life, or as a preparation for higher mathematical and other studies.

NOTATION;

Or, the Art of Writing Numbers in Figures.

6. For this purpose there are ten several forms, or figures, in use amongst us. And with these ten figures only, but differently placed, and, as occasion may require, repeated, can any number, however large, be most clearly and conveniently expressed.

7. The ten Figures are as follow. The value, or amount, which each of them serves to express, is written underneath, in words.

1 2 3 4 5 6 7 8. 9 0

One, two, three, four, five, six, seven, eight, nine, nought, or cypher.

8. Of the last figure, called nought, or cypher; its name, nought, comes from the Saxon word, nothing; and nothing does it express; that is to say, nothing in amount. However, it is useful; and, indeed, it cannot be dispensed with, when we come to write down the higher numbers, as will appear hereafter.

9. Of the other nine figures, be it observed, that each of them, when standing alone, and unconnected with any of the others; or, when standing the last, on the right hand, of a series of figures; each of these nine, when so standing, expresses merely the number which we have written underneath it: but the highest, even, of these figures, in this its single and separate state, expresses a number no higher than nine; and, yet, as the learner knows, hundreds, thousands, and yet higher numbers, are continually required to be expressed.

10. To do this, that is, to express large numbers, by figures, each figure differing in form from every other figure, and, thus to have a different form for every different number; to do this, would be an endless, and an impracticable task. So, another, and a very convenient and neat mode has been adopted for accomplishing this purpose; of which mode we now proceed to treat.

11. The manner, then, 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.

12. This change of place, however, of which we have to speak, is not a mere removal of the figure, or figures; for a mere removal, as has before been said, 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.

13. In order to fix in the mind of the young student the effect of this changing of 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 student take one of them, the first, for instance, and place it right before him on the table: thus standing, singly, alone, the figure expresses the number, one. Then let him take another of the figures, (the second, let it be) and place it over, or under, or to the left of the first, and this first figure still expresses only one; its value is still unaffected by the approach of the other