to study, as both useful and agreeable; useful, because it would show him the means of accounting to himself for the result of his own labours; and agreeable, because it would afford him a pleasing object of speculation for his winter evenings. should be delighted to see several such lads, passing an evening together, with this book between them, each his slate and pencil before him, discussing, mutually giving and solving, the questions which they learn from it to form out of the occurrences around them. I can promise them more satisfaction from it, than in their passing that time in the bar-room of a public house, or a grocery; and more beneficial, economical results, from the expenditure in book, slate, and pencil, to assist their studies, (for they must write every thing,) than were they to lay out the cost in the vile liquor, that emptiness of mind leads them to call for; they will soon be able to calculate: that they even make a saving, if they write their full studies, ideas, and questions, on paper, with pen and ink, in comparison with the expences of the deleterious pleasures of a bar-room. If I should succeed only in this part of my aim, I would consider my labour as sufficiently rewarded; and I would have the greatest enjoyment, to meet with such a company, afford them assistance, and partake of their rational amusement.

For the use of this book, I should like to advise, the teacher, as well as the student, first to peruse attentively the theoretical principles of any rule or subject, and then exercise his scholars, or himself, in the application, which will give him an opportunity to generalise, and clear up their, or his, ideas properly; and after having gone through any of the principal subdivisions, to take a general view of the whole; taking care to comprehend the leading principles, and the mode of considering the subject, that has been treated of; in this way he will be enabled

to make a proper use of it in the parts to be treated

next.

It is an unavoidable condition in every systematic work, that the subsequent parts shall be grounded upon the preceding ones, and therefore these must be supposed known in the progress of the work, as it proceeds. Therefore also the study of no systematic and good work, can be begun in any other part than at the beginning, by any scholar; that is, a person not fully acquainted with the whole subject of the book, but seeking instruction from it. If any person thinks he knows already some of the elementary parts, and wishes to study only the subsequent part, it is necessary for him to read over, attentively, the parts with which he is acquainted; to make himself acquainted with the manner in which the author expresses himself upon those subjects, which he has his own ideas upon. By comparing these together, he will be able to understand properly, afterwards, those parts with which he is not acquainted; and therefore read and study with success; which otherwise will certainly not be the case. This is nothing else but what is necessary between all men, in any intercourse, that is, the necessity of being acquainted with each others' language.

New-York, October, 1826.

F. R. HASSLER.

PART IM

FIRST ELEMENTS AND DEDUCTION OF THE FOUR RULES OF ARITHMETIC.

CHAPTER I.

Fundamental Ideas of Quantity.-System of

Numeration.

1. QUANTITY, which is the object of Arithmetic, is the idea that has reference to any thing whatever, arising from the consideration of its being susceptible of being more or less; without regard to the nature or kind of the thing itself. It is not therefore an absolute existence; but a relative idea, that can be referred to any object whatever.

§ 2. No quantity therefore can be called great or small, much or little, in itself; it can be so only in relation to another quantity of the same kind, which would be smaller or greater.

§ 3. Objects of different kinds cannot be compared with each other directly by their quantity only. When therefore Objects of different kinds are to be considered in Arithmetic, it becomes necessary: that a certain relation be given between them, which is completely arbitrary as to quantity itself, and must be determined before any comparison can take place.

§ 4. The mutual relation of quantities to each other, under certain given conditions, is the object of arithmetic. In this general acception then it admits any number of systems of combination, that the imagination can devise.

5. To form a clear and distinct idea of arithmetic it is necessary, to impress the mind fully with these fundamental ideas, and the general principles that follow from them. By comparing every operation of arithmetic with them, they will become always more and more clear and useful; the whole system of arithmetic will become the more simple, the more its principles are generalized.

6. Common arithmetic, which might also be called with propriety, determinate arithmetic, limits itself to the most simple combinations of quantities, and these are all grounded successively upon the first elementary idea of increase or decrease, or more or less, either simple or repeated successively, or according to certain determined laws.

§ 7. To express quantities we make use, in our system of common arithmetic, of ten figures only, by the means of which, and by their relative places, according to a certain law, we can express any quantity whatsoever. This law is called the system of numeration; and in particular the decimal system, from the individual circumstance, of its using ten different figures, nine of which are significant, and the tenth indicates the absence of the quantity (or thing, or object.)

§ 8. These figures are in regular succession 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0; this last is used to denote the absence of a quantity; the 1, denotes the unit of any object, of whatever kind or nature it may be; the subsequent denote in regular succession each one object more than the one before it.

§ 9. To denote quantities which exceed the number of significant figures, (or above 9,) recourse is had to a law that assigns superior values to these figures, according to the order in which they are placed, assigning to them a value as many times greater, in every successive change of place from the right to the left, as the number of figures indicates,