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PREFACE

TO TIIE THIRD EDITION.

WHEN a new york is offered to the public, especially on a subject 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 new; and, as to the second, a cart did 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 froni the preceding?' The answer is, Yes, in many respects. The present edition proiesses to be strictly on the Pestalozzian, or induetive, 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 lo consist in adhering more closely to the true spirit of the Pestalozzian plan; consequently, in difering froin other systems, it differa less from the Pestalozzian. This similarity will now be shown.

1. The Pestalozzian professes to unite a complete system of Mental with Written Arithm.ctic. So does this.

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

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

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

The following are a few of the prominent characteristics of this work, in which it is thought lo 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 Felloroship, &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 pointe then the aggregate of these results, with points, for an ansu by which the relative value of the whole is determined, 1 furnishing a complete test of the knowledge of the pupil.

On

is a characteristic difference between this and the former editions.

5 A new rule for calculating interest for days with months.

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, previously and subsequently to their insertion.

9. As this mode of teaching recognizes 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 accom. pany this work.

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

As, in this work, the common rules r[ Arithmetic are retained, perhapg the reader is ready to propose a question frequently asked, "What is the use of so many rules?» “Why not proscribe then?" The reader must nere be reminded, that these rules are taught differently, in this systern, 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 sunimed in the form of a rule, wiichi, for convenience sake, is nained. Is there any improprieiy in this? the contrary, is there not a greai convenience in it? Should the pupil he left to form his own rules, it is more than robable he might noistake the most concise and practical one. Besides, diferent minds view things differently, and draw different conclusions. Is there no beneiil, 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 gene so far as io condemn the Rule of Three, or Propor. tion, and almost all the successive rules growing olit of it.

With mors reason, they mighi condemn Long Bivision and even Short Division ; and, in fact, all the common and fundamental rules of Arithmetic, except Adili tion ; for tliese my all be iraced to that. The only question, then, is, “To what extent shall we go :") To this it is replieri, As far as convenience requires. As the Rule of Three is generally taught, it must lie confessed, that almost any thing else, provided the nuns of the pupil ne exercisel, would be a good substitute. But when taught as it should be, and the scholar is led on in ihe saine train of thouunit that originated the rule, and thus effectually made to see that it is simply a convenient method of arriving at the result of both Mulielication and Division combined, its necessity may be advocated with as muen reason as any fundamental rule. As laŭgue in this work, it actually sa ves more figures ilian Short, compared wili Long Divisiuni. Here. then on the ground of convenience, it woud be reasonable in infer. thal jis retention was more necessary than either. But, waiving its tiility in this respect, there is another view to be taken of this suije(t, and it not the least in importance, viz. the ideas of beauty arising froin viewing the harmonious relations of numbers. Here is a delightul lipid for an muistive mind. It liere imbibes truths as lasting as life. Whea thic utiliis and convenience of this rule are once conceded, ali the other rules growing out of this will demand a place, and or the same reason.

It may, perhaps, be asked by many, " Why not take the principle with. wut the name?" To this it is aya ili replieit, Convenience forbids. The name, the pupil will see, is only an aggregate term, given to a process in

dying several distinct principles. Anu is there no convenience in tliiss Shall the pupil, when in actual business, be obliged to call off his mind from all other pursuits, lo trace a train of deductions arising from abstract reasoning, when his attention is most needed on other subjects? With ag 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, although not quite so conveniently With these remarks, the subject will be disinissed, merely adding, by way ef request, that the reader will defer his decision till he bas examined the doctrine of Proportion, Fellowship, &c., as taught in this work.

Tbe 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 rules, has long since been settled; the only ques tion is, with regard to the expediency of introducing 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 truths, 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 improvemeot of the mind, and the practical utility which ihese rules atord ? " Besides, there always will, in some nook or other, spring up some poor joy of mathematical genius, who will be dem sirous of extending his researchies 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 lotters?

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 mind by the acquisition of useful and practical knowledge, which may be in the possession of alınost every learner, as to reserve this interesting portion of Mathematics for a favored few, and, in the mean time, to divert the attention of the pupil to less useful subjects ?

Tile 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 ihe reason why his product, in decimals, should be less than either factor; as, for instance, why the solid contents of a half an inch cube si:ould 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 a solid inch.

In this work, the author has endeavored to make every part conform to this maxiin, viz. THAT NAMES SHOULD SUCCEED IDEAS

ibis method of communicating kno!ledge is diametrically opposed to that which obtains, in inany places, at the present day. The former, by first giving ideas, allures the punit into a luminous comprehension of the subject, while the latter astomas lin. at first, with a pompous name, to which he seldom affixes any definite idens, and it is exceedingly problematical Wiether he ever will. In adılition to ilis is the fact, that, by the last-mentioned meth. od, when the name is given and the process shown, not a sivgle reason of any operation is ancos: but the pupil is dogmatically told he must proCted thus and so, and he will come oilt so and so. This mode of teaching is very much as if a merchant of this city should direct his elerk, withcut intrusting him with my business, first io go to Seuth Boston, then to the stale-house, afterwards to the market, and chen to return, leaving him to surmise, ir'ize can, the cause of all this peregrinaiion. Many aro fools enough to take this jaunt pleasantly; others are restiff, and some fractinus. This sentiment is that sustained by an article in Miss Edg worth's works. fium which the following exiract is made: “Ach seeming stupidity, in learning arithmetic, may, perhaps, be a proo!

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louigence and good senge. It is easy to make a boy, who dnes nak masa, repeal, by rote, any techuical rules, which a coinmon writing master, withi magisterial solemnity, inay lay down for him; but a child who reasons will not be thus easily managed; he stops, frowns, besitates, questions his mas ter, is wretched and refractory, until he can discover why he is to proceed in such and such a manner; he is not content with seeing his preceptor make figures and lines on the slate, and perform wondrous operatiuns with the self-complacent dexterity of a conjurer; he is not conteni to be led to the treasures of science blindfold; he would tear the bandage from his eyes, that he might know the way to them again.”

In confirmation of the preceding remarks, and as fully expressive of the author's views on this subject, 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 favored sister, GEOMETRY, either in precision of ideas, in clearness and certainty of deinonstration, in practical utility, or in the sublime deductions of the most interesting truths.

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

THE AUTHOR.

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...................

......................62

Division of Feileral Money,....

..113

To multiply by 1, 1,5., &C...., ................................65

Practice in Ferleral Money,...

..68

(questions on the foregoing --Bills of goods solil,.

.69

Reduction-Tables of Money, Height, Measure, &c.,............. .71

Compound Addition,......

Compound Subtraction,

** ...............92

Compound Multiplication,...

100

Compound Division,

Questions on the furegoing,

...97

.....103

...104

........

.......ros

FRACTIONS.

COMMON, or VULGAR,.

Fractions arise from Division,..

..........................................106

Proper, Improper, &e.,....

I'o change an Improper Fraction to a Whole or Mixed Number,........... 109

To change a Mixed Number to an Improper Fraction,........

...........11

t'u reduce a Fraction to its lowest Terms......

........108

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