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When a new work is offered to the public, especially on a stbject abounding with treatises, like this, the inquiry is very naturally made,“ Does this work contain any thing new?" " Are there not a hundred others as good as this?". To the first inquiry it is replied, that there are many things which are believed to be now; and, as to the second, a candid public, after a careful examination of its contents, and not till then, it is hoped, must decide. Another inquiry may still be made : "Is this edition different from the preceding?" The answer is, Yes, in many respects. The present edition professes to be strictly on the Pestalozzian, or inductive plan of teaching. This, however, is not claimed as a novelty. In this respect, it resembles many other systems. The novelty of this work will be found to consist in adhering more closely to the true spirit of the Pestalozzian plan ; consequently, in differing from other systems, it differs less from the Pestalozzian. This similarity will now be shown.

1. The Pestalozzian professes to unite a complete system of mental with written arithmetic. So does this.

2. That rejects no rules, but simply illustrates them by men. tal questions. So does this.

3. That commences roith examples for children as simple as this, is as extensive, and ends with questions adapted to minds as mature.

Here it may be asked, “In what respect, then, is this different from that?" To this question it is answered, In the execution of our common plan.

The following are a few of ihe prominent characteristics of this work, in which it is thought to differ from all others.

1. The interrogative system is generally adopted throughout this work.

2. The common rules of arithmetic are exhibited so as to correspond with the occurrences in actual business. Under this head is reckoned the application of Ratio to practical purposes, Fellowship, &-c.

3. There is a constant recapitulation of the subject attended to, styled Questions on the foregoing.

4. The mode of giving the individual results without points, then the aggregate of these results, with points, for an answer, by which the relative value of the whole is determined, thy furnishing a complete test of the knowledge of the pupil. This is a characteristic difference between this and the former editions.

5. A new rule for calculating interest for days with month a

6. The mode of introducing and conducting the subject of Proportion.

7. The adoption of the federal coin, to the exclusion of sterling money, except by itself.

8. The arithmetical tables are practically illustrated, prediously and subsequently to their insertion.

9. As this mode of teaching recognises no authority but that of reason, it was found necessary to illustrate the rule for the extraction of the cube root, by means of blocks, which accompany this work.

These are some of the predominant traits of this work. Others might be mentioned, but, by the examination of these, the reader will be qualified to de cide on their comparative value.

As, in this work, the common rules of arithmetic are retained, perhaps the reader is ready to propose a question frequently asked, "What is the use of so ma ny rules?” “Why not proscribe them?". The reader must here be reminded, that these rules are taught differently, in this system, from the common method. The pupil is first to satisfy himself of the truth of several distinct mathematical principles. These deductions, or truths, are then generalized; that is, briefly summed in the form of a rule, which, for convenience" sake, is named. Is there any impropriety in this? On the contrary, is there not a great convenience in it? Should the pupil be left to form his own rules, it is more than probablo he might mistake the most concise and practical one. Besides, different minds view things differently, and draw different conclusions. Is there no benefit, then, in helping the pupil to the most concise and practical method of solving the various problems incident to a business life?

Some have even gone so far as to condemn the Rule of Three, or Proportion, and almost all the successive rules growing out of it. With more reason, they might condemn Long Division, and even Short Division; and, in fact, all the common and fundamental rules of arithmetic, except Addition ; for these may all be traced to that. The only question then is, "To what extent shall we go?" To this it is replied, As far as convenience requires. As the Rule of Three is generally taught, it must be confessed, that almost any thing else, provided the mind of the pupil be exercised, would be a good substitute. But when taught as it should be, and the scholar is led on in the same train of thought that originated the rule, and thus effectually made to see, that it is simply a convenient method of arriving at the result of both: Multiplication and Division combined, its necessity may be advocated with as much reason as any fundamental rule. As taught in this work, it actually saves more figures than Short, compared with Long Division. Here, then, on the ground of convenience, it would be reasonable to infer, that its retention was more necessary than either.. Buty waiving its utility in this respect, there is another view to be taken of this subject, and that not the least in importance, viz. the ideas of beauty arising from viewing the harmonious relations of numbers. Here is a delightful field for an inquisitive mind. It here imbibes truths as lasting as life. When the utility and convenience of this rule are once conceded, all the other rules growing out of this will demand a place, and for the same reason.

It may, perhaps, be asked by many, "Why not take the principle without the name. To this it is again replied, Convenience forbids. The name, the pupil will see, is only an aggregate term, given to a process imbodying several distinct principles. And is there no convenience in this! Shall the pupil, when in actual business, be obliged to call off his mind from all other pursuits, to trace a train of deductions arising from abstract reasoning, when his attention is most needed on other subjects. With as much propriety the name of captain may be dispensed with ; for, although the general, by merely summoning his captain, may summon 100 men, still he might call on each separately, elthough not quite so convenicntly With theso remarks, the subject will be

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dismissed, merely adding, by way of request, that the reader will defer his decision till he has examined the doctrine of Proportion, Fellowship, &c., as taught in this work.

The APPENDIX contains many useful rules, although a knowledge of these is not absolutely essential to the more common purposes of life. Under this head are reckoned Alligation, Roots, Progression, Permutation, Annuities, &c. The propriety of scholars becoming acquainted, some time or other, with these

Fules, has long since been settled; the only question is, with regard to the ex3pediency of introdacing them into our arithmetics, and not reserving them for

our algebras. In reply to this, the writer would ask, whether it can be supposed, the developement of these traths, by figures, will invigorate, strengthen, and expand the mind less than by letters? Is not a more extensive knowledge

of the power of figures desirable, aside from the improvement of the mind, and *the practical utility which these rules afford? Besides, there always will, in

some nook or other, spring op some poor boy of mathematical genius, who will be desirous of extending his researches to more abstruse subjects. Must he, as well as all others, be taxed with an additional expense to procure a system, containing the same principles, only for the sake of discovering them by letters?

Position, perhaps, may be said to be entirely useless. The same may be said of the doctrine of Equations by algebra. If the former be taught rationally, what great superiority can be claimed for the one over the other?

Is it not obvious, then, that it is as beneficial to the pupil to discipline his al mind by the acquisition of useful and practical knowledge, which may be in

the possession of almost every learner, as to reserve this interesting portion of mathematics for a favoured few, and, in the mean time, to divert the attention of the pupil to less useful subjects?

The blocks, illustrative of the rule for the Cube Root, will satisfactorily account for many results in other rules ; as, for instance, in Decimals, Mensuration, &c., which the pupil, by any other means, might fail to perceive. By observing these, he will see the reason why his product, in decimals, should be tess than

either factor i a3, for instance, why the solid contents of a half an inch cabe should be less than half as much as an inch cube. In this case, the factors are each half an inch, but the solid contents are much less than half & solid inch.

In this work, the author has endeavoured to make every part conform to. this maxim, viz. THAT NAMES SHOULD SuccEED IDEAS. This method of communicating knowledge is diametrically opposed to that which obtains, in many places, at the present day. The former, by first giving ideas, allures the pupil into a luminous comprehension of the subject; while the latter astounds him, at first, with a pompous name, to which he seldom affixes any definite ideas, and it is exceedingly problematical whether ho ever will. In addition to this is the fact, that, by the last mentioned method, when the name is given and the process shown, not a single reason of any operation is adduced; bat the pupil is dogmatically told he must proceed thus and so, and he will come out so and so. This mode of teaching is very much as if a merchant of this city should direct his clerk, without intrusting him with any businoss, first to go to South Boston, then to the state-house, afterwards to the market, and then to return, leaving him to surmise, if he can, the cause of all this peregrination. Many are fools enough to take this jaunt pleasantly; others are restiff, and some fractious. This sentiment is fully sustained by an article in Miss Edgeworth's works, from which the following extract is made: “A

child's seeming stapidity, in learning arithmetic, may, perhaps, be a proof of di intelligence and good sense, It is easy to make a boy, who does not reason, .

repeat, by rote, any technical rules, which a common writing master, with magisterial solemnity, may lay down for him ; but a child who reasong will not be thus easily manager; he stops, frowns, hesitates, questions his master, is

wretched and refractory, until he can discover why he is to proceed in such viss and such a manner; he is not content with seeing his preceptor make figures od rand lines on the slate, and perform wondrous operations with the self-com

placent dexterity of a conjurer ; he is not content to be led to the treasures of

science blindfold; he would tear the bandage from his eyes, that he might knon the way to them again."

In confirmation of the preceding remarks, and as fully expressive of the author's views on this subjeet, the following quotation is taken from the preface to Pestalozzi's system.

“'The PestALOZZIan plan of teaching ARITHMETIC, as one of the great branches of the mathematics, when communicated to children upon the principles detailed in the following pages, needs not fear a comparison with her more favoured sister, GEOMETRY, either in precision of ideas, in clearness and certainty of demonstration, in practical utility, or in the sublime deductions of the most interesting truths.

“In the regular order of instruction, arithmetic ought to take precedence of geometry, as it has a more immediate connexion with it than some are willing to admit. It is the science which the mind makes use of in measuring all things that are capable of augmentation or diminution ; and, when rationally taught, affords to the youthful mind the most advantageous exercise of its reasoning powers, and that for which the human intellect becomes early ripe, while the more advanced parts of it may try the energies of the most vigorous and matured understanding." January, 1829


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