INTRODUCTION.

I. NUMBER is one of the early subjects of attention. It is impossible for any one to be entirely ignorant of it, after he is capable of contemplating the succession of objects around him. LACROIX justly observes, that "by comparing the objects which come within the reach of our senses, we perceive they have magnitude. When they appear in the form of a collection of similar things, or distinct parts, the term number is applied; and numbers in particular are the objects of Arithmetic.

Few sciences are so constantly necessary as this for the purpose of intelligibly communicating ideas, or the ready transaction of business. Number enters into almost every subject, whether of philosophy or religion,—and he, who would either receive or communicate instruction, must have some of its principles constantly in mind.

But, while it needs no argument to establish the claims of this science upon the attention of the young, it needs little time to prove that it may be pursued in such a way as to furnish very little improvement to them. Such evidence is furnished fully to every one, who has taken pains to examine any considerable number of those, who, having ciphered through" their book, suppose themselves very well acquainted with the whole science. The natural consequence of the manner adopted by many teachers, and the character of the books used by many of the members of our schools, could not well lead to different results. Little more has been attempted than to commit rules to memory and then perform operations, according to the directions therein given. Ability to obtain the answer" has satisfied the scholar, and he is willing to think his knowledge complete. After all, however, in a majority of cases, he has no more knowledge of the principles-the reasons-why the rule furnishes the given directions, than he would have of a proposition in Euclid, written in a language he does not understand. And, indeed, how should he, when in general the text-book with which he is furnished is without demonstra

tions, and his teacher is well satisfied if the rule is recited and the operation carried through in the manner directed?

Ask a scholar of this description, whether he can give you a true answer to a sum, and he is ready to reply, at once, in the affirmative, if the rule to do it is specified. But if he is directed to ascertain the rule for himself, his resort is to the arithmetic, and if the sum corresponds in character to those, which he finds there under any particular rule, he tries to perform it by the aid of that rule. If in this way he obtains a satisfactory result, he is at once, in his own opinion, a master of the science. If however, unsuccessful, he tries another rule-and if this will not do, another-till presently, discouraged in his ill-directed effort, he abandons it in despair, and is ready to say "it can't be done."

Another prominent fault, which must have been noticed by every careful observer, is this-the knowledge thus acquired is retained but a very short time. Few, unless engaged in business, which constantly requires their use, retain any considerable degree of their former arithmetical acquisitions -save that of the fundamental rules and interest, perhaps for any great length of time. The practice is discontinued, and therefore nearly all is lost. What benefit has been derived, then, from all the time devoted to the subject in the district school or elsewhere, year after year?

But let the principles on which every rule is founded, be thoroughly understood; let a portion of the time spent in the mere practical part, be devoted to acquiring a thorough acquaintance with the reasons on which rules are founded, and we have every reason to apprehend these principles will remain in the memory, when the details shall have been forgotten. Rules may be originated for himself, by any one, who is able to look over the whole ground-and when, from the conditions of the question proposed for solution, the learner is able to form a correct rule, he possesses that kind of knowledge, which is adapted to the wants and exigencies of life. He, who forms the key, can surely apply it, and unlock the door at his will.

It has been asserted by some, that the time allowed to the young, is merely adequate to the full acquisition of the practical part of arithmetic, and hence, the reason why the theory should be nearly disregarded. In reply, we beg leave to say, we have yet to learn that the right way is longer than the wrong, and that a path, lighted at every step, requires