have in early youth been taught by the negligent method of mere rules, and have at a later period attained scientific eminence, to decide between this and any contrary assertion.

The difficulties that the young experience on entering upon any scientific studies, in colleges, or otherwise, are well known; the path to be followed there, must be that of reasoning, and no preparations are made for this by their previous education; for the cultivation of the memory alone, is, from the very constitution of the human mind, always detrimental to the reasoning faculty.

However the opportunity, as has been stated, exists, of cultivating the reasoning faculty at an earlier period, by familiarizing the scholar with the simple reasonings of elementary arithmetic. The step from that to higher or general arithmetic, usually called Algebra, becomes by this mode, both short and simple, as in its nature it really is; and the scholar who does not wish to go farther than common arithmetic, can alone obtain the knowledge of the propriety or principles of its application to any occurrence in common life, by a knowledge of it, founded upon correct reasoning. It is entirely wrong to say and act, upon the ground, "I want to know how to do this or that," the principle must be, "I wish to understand this or that," if ever any lasting good result shall be obtained.

My object in undertaking this work was not to swell the number of elementary treatises on arithmetic, but may be stated as follows.

1st. I wish to smooth the path of the teacher and the scholar, by explaining and proving, the propriety and correctness of any step that is taken, by previous reasonings, leading to the discovery of the principle that ought to direct it, and therefore pointing out the rule for the appropriate operation; and I have, therefore, not been content to give the final

result alone, and the example for its proof, which is an individual, and consequently a defective proof, while reasoning always leads to general propositions and proofs. In this way we attain, step by step, to the real scientific structure of this elementary science, and thus all the operations become satisfactory to the mind, and therefore agreeable to the growing intellect of the scholar.

In carrying such a system through the whole extent, to that point where more general and extensive considerations, of a higher analytic nature, are to guide us, I have even thought it possible to make a treatise, which a man of science might look at with some satisfaction, and by which the young scholar would arrive at the entrance of his higher scientific studies, properly prepared by a correct habit of reasoning.

2d. The young and untutored mind, in truth, reasons analytically; a boy, and in fact a man, asks always WHY; and as he enters more and more deeply into the investigation, continues to ask the reason of every thing that is said to him, in the way of explanation. The reason of this lies in the nature of his situation; he cannot proceed synthetically, because synthesis needs some previous data, averred, given, or adopted, on which to build the reasoning to arrive at a conclusion. This does not yet exist at this early stage of instruction.

In following this mode, and grounding every conclusion upon inquiry, of which the ground lies, either in the human mind itself, even untutored, or in the result of preceding investigations, I intend to make a book which a lad remote from cities, although he might not have had the benefit of a good early education, can take in hand usefully, and which a simple knowledge of reading, coupled with his own desire for improvement and instruction, would induce him to take up, and undertake

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. I 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, upon which he has his own ideas. 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.