VIMU AIMBOTILIAO

QA102 H37

SOUTHERN DISTRICT OF NEW-YORK, SS.

BE IT REMEMBERED, That on the 6th day of OcL.S. tober, A. D. 1826, in the 51st year of the Independence of the United States of America, F. R. HASSLER, of the said District, hath deposited in this office the title of a Book, the right whereof he claims as Author, in the words following, to wit:

Elements of Arithmetic, Theoretical and Practical; adapted to the use of Schools, and to Private Study. By F. R. HASSLER, F.A. P. S.

In conformity to the Act of Congress of the United States, entitled "An Act for the encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the authors and proprietors of such copies, during the time therein mentioned." And also to an Act, entitled "An Act, supplementary to an Act, entitled an Act for the encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the authors and proprietors of such copies, during the times therein mentioned, and extending the benefits thereof to the arts of designing, engraving, and etching historical and other prints."

JAMES DILL,

Clerk of the Southern District of New-York.

CAJORI

INTRODUCTION.

ARITHMETIC contains the first elements of reasoning upon quantity; its principles take their rise in ideas so simple as to be adapted to the most untutored mind, and to the lowest capacity. It is at the same time so indispensable for every human being, not only in common life, but in the pursuits of the highest sciences, that it forms the most proper, and has always formed one of the principal branches of the earlier education of youth.

By its very nature it furnishes the means of developing the reasoning faculties, from the time of their first beginning to expand themselves, and of habituating them to correctness and precision. It therefore gives the human mind the power and disposition to reason upon sound and correct principles.

It is therefore the duty of the faithful teacher of youth, (not the mere teacher for his own private emolument,) to take advantage of this property of arithmetic, and apply it to cultivate the mind, and enlighten the understanding of his scholars, by a proper reasoning in this elementary science; he should not make it the object of the memory alone; a method that leaves no impression upon the mind, whose results are therefore lost again as soon as the school is dismissed.

To neglect to take this advantage of the study of arithmetic, is either a proof of ignorance, or an actual dereliction of duty. This may appear strong to many people, but strength is the essential property of truth. I can safely appeal to those who 918171

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 method, 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