A Treatise on Arithmetic

PREFACE.

A preface is not unfrequently made the vehicle of conveying to the public an author's apology, for having presumed to think and write upon a subject, on which many others have thought and written before him. Such an apology, however, while it betrays a want of that manly confidence, with which every one, who is conscious of having striven, with integrity of purpose, to promote the good of his fellows, should look forward to the award of an intelligent public; is in itself, unnecessary, because ineffectual to accomplish the object, for which it is intended. In the present enlightened age, works are judged of, by their own intrinsic merit, and not by what their authors may choose to say in their favor: and nothing can be more vain, than to attempt, by fair pretences, to palm off upon the public an inferior production. Influenced by considerations like these, the author of the following treatise was about to lay it before the public, without apology, and with hardly a prefatory remark; confident, that if it possess merit, this will be discovered and approved; i otherwise, that whatever he could say, would be a mere waste of breath.

He deems it due, however, to those, whose interesting task it is, "to rear the tender thought," to state in what respects he has chosen to differ from other arithmeticians, and also, briefly to assign his reasons for so doing. These are the following:

1. The book commences in a style likely to attract and fix the attention of the pupil, and which, for its simplicity, is calculated to prevent the prejudices, which the young are so apt to conceive against mathematical studies. It is believed that, by commencing below the level of the pupil's capacity, and thus rendering it certain that every thing is understood in the outset, important advantages are gained; among which it is not the least, that the learner becomes encouraged to expect that he shall understand the parts which succeed.

2. In passing from the mental to the written exercises, it will be observed that the transition is so gradually made, as to render the resemblances and the differences entirely distinct. While the same general principle is seen obviously to pervade the whole of the same class of examples, the facilities of calculation afforded by the scheme of notation, which are of course mostly peculiar to written arithmetic, may easily be distinguished.

3. In developing the principles of those operations, embraced under what are usually called the ground rules, it has been on object to illustrate each step of the process separately. This is an important distinction between the present and all former treatises on the subject. Thus, in addition, the first examples only differ from the preceding mental exercises, in requiring the numbers concerned to

be written. In the beginning, a single order, only, is employed; af terwards, two, three, and so on, the greatest simplicity being still preserved, in order that the mind of the pupil may be occupied with nothing, except the mere arrangement of his numbers. Afterwards, larger numbers are used, containing many orders, but so contrived, that there shall be no necessity of carrying from one order to another. Finally, the pupil is taught to carry, by means of a series of examples, commencing again with the greatest simplicity, and ad. vancing to the more difficult combinations. The learner is at first, however, required actually to set down the number carried, until he cannot fail to perceive the advantage of adding it mentally. All these things are usually explained in a single example, and that often consisting of numbers so great that the learner can have no conception of their amount, nor, of course, of the principles concerned in the process. The consequence is, that the young begin. ner, perplexed by the number and variety of things which he is required to remember without comprehending, and tired of the irksome task of performing operations which are to him only not com. pletely mechanical, becomes disgusted with the study, and imbibes prejudices against all kinds of mathematical pursuits, which continue to the end of his life.

4. This treatise is much more full than any designed for the same purpose which has preceded it. A glance at the table of contents, will be sufficient to show, that the variety of subjects treated of, is much greater than is usual, and that much useful information is given on numerous topics, collaterally connected with the main subject.

5. The abbreviations in calculation are thought to be a valuable addition to a practical work on this subject. These abbreviations will often furnish a convenient mode of verification or proof of an operation performed by the common method.

6. The articles on circulating decim ls, are believed to contain much which will interest. It is delightful to contemplate the beauty and harmony of numbers, as they are exemplified in the perfect conformity even of this apparently anomalous class of quantities, to fixed and unchangeable laws. No book, before the present, has given a popular exposition of the subject, and it is thought that in this part of the work, even teachers may in some instances meet with new ideas.

7. INTEREST is treated with the fulness which its importance demands. The method by decimal multipliers, which is becoming so deservedly popular from its simplicity, has been adopted and minutely explained.

8. The author is inclined to think that his method of developing the principles of ratio and proportion, will meet approbation. It is certain, that the subject of compound proportion, can no where be found treated with equal simplicity. The process by means of a single fraction, is likewise original, and is exceedingly concise. The author has employed this process with his classes, and has found it eminently successful. Pupils learn it with great facility, and when it is once familiar, they solve by means of it, the most complex questions, almost as easily as the most simple.

9. Of the extraction of the square and cube roots, an analytic investigation is here, for the first time, given. By this means, it is hoped, that this difficult subject will be rendered more intelligible.

10. In general, there is a simplicity in the explanations and illustrations which it is believed will not be found elsewhere. This arises from the fact that they are the result of EXPERIMENT, upon minds of every degree of native acuteness, and energy, and in every stage of cultivation.

11. By the arrangement, TECHNICAL TERMS are introduced only after the pupil has been made thoroughly acquainted with the processes in which it is necessary to apply them, or with the circumstances which render it proper that they should be employed. We all know how difficult it is to form correct ideas, even of sensible objects, from description merely; and it is by no means surprising, that when the subject of a description is an abstract term, or some one of the numerous technicalities of science, it should be impossible for a child to form any idea of what is intended by the language. But when a name is applied to something already familiar, the ideas attached to it are perfectly distinct and definite. It was in a similar manner that the child first learned his mother tongue, and it is in this way that we ourselves are from time to time, adding to our stock of language.

12. By a similar arrangement, the rules are made to succeed the knowledge of the processes, which they describe. For mere practice, therefore, they are unnecessary, and indeed, they are not intended, except in a very few instances, to serve as the learner's guides. They are merely given, as furnishing concise and accurate language for the expression of ideas supposed to be previously familiar.

13. The rules will be found more brief and more easy to be committed in this book than in any which the author has seen. It has been a prominent object to make them so. And since the pupil is not expected, and ought not to be suffered to depend on them as a guide, conciseness has been attainable, without the sacrifice of any desirable end.

These constitute the prominent points of difference between the present and former treatises on this subject. In no one of these par. ticulars has any deviation been made from other writers, without long and patient observation of the effect of different methods of instruction upon the youthful mind. This is the test to which the author has subjected all his proposed improvements, and the uniform success which has attended his experiments, leads him to look with confidence for the approbation of the public.

HARTFORD, JULY 30, 1830.