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Shall the pupil, when in actual business, be obliged to call off his mind rom 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, although not quite so conveniently With these remarks, the subject will be 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 rules, has long since been settled; the only question 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 developenient 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 figa ures 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 up some poor boy of mathematical genius, who will be desirous of extending his researches to more abstruse subjects. Must he, as weli 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 ra: tionally, what great superiority can be claimed for the one over the other? Is it not obvious, then, that it is as heneficial to the pupil to discipline his mind by the acquisition of useful and practical knowledge, which may fie in the possession of almost every learner, as to reserve this interesting portion of Mathematics for a favored few, and, in the mean time, to divert ihe 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, hy any other means, might fail to perceive. By observing these, he will see the reason why his produci, in decimals, should be less than either factor; as, for instance, why the solid contents of a half an inch cube 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 a solid inch.

In this work, the author has endeavored 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 arfixes any definite ideas, and it is exceedingly problematical whether be ever wili. In addition to this is the fact, that, by the last-mentioned meth od, when the name is given and the process shown, not a single reason of any operation is adduced; but 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 it a merchant of this city should direct his clerk, without intrusting him with any business, 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 Edge Wurth's works, from which the following extract is made : “A childers seeming stupidity, in learning arithmetic, may, perhaps, be a proof of in

telligence and good sense. It is easy to make a boy, who does not reason, repeat, by, rote, any technica) rules, which a common writing master, with magisterial solemnity, may lay down for him ; but a child who reasons will not be thus easily managed; he stops, frowns, hesitates, questions his master, 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 operations 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 cyes, that he migbt 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 sistar, 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 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 becoines early ripe, while the more advanced parts of it njay try the energies of the inest vigorous and matured understanding."

THE AUTHOR January, 1829

....152

....171

...182

.......183

...184

...189

Reduction of Currencies,

Questions on the foregoing,...

....160

Interest,...

....163

Concise Rule for calculating Interest in New York State,...

.170

Commission, Insurance, Stock, Loss and Gain,...

Time, Rate per cent., and Amount, given, to find the Principal,.. ...173

Time, Rate per cent., and Interest, being given, to find the Principal,......176

The Principal, Interest, and Time, being given, to find the Rate per cent.,.. 176

The Principal, Rate per cent., and Interest, being given, to find the Time,..178

Compound Interest--Compound Interest by Table,

,179

Equation of Payments,..

Questions on the foregoing,.

Rule of Three, by Analysis,

Ratio, or the Relation of Numbers,..

Proportion arising from Ratio, or from Multiplication and Division, ....

189

Application of Ratio by Rule......

..192

Rule of Three in Vulgar and Decimal Fractions,.

..196

Compound Proportion, ur Double Rule of Three, by Ratio and Analysis,...198

Questions on the foregoing,...

To compute the Interest on Notes with Endorsements-three modes,......207

Practice in. Con poun' Numbers,......

..211

Fellowship - by Analy vis-by Ratio,..

....213

Compound Fire wehip. -by Analysisby Ratio,..

....216

Mensuration--Square Measure,...

....217

Solid, or Cubi? N'easui ?,,

..221

Duodecimals—. Vu 'tiplic'ition of Duodecimals,...

.225, 2

Questions on th 1 vrego. ng...

.223

To calculate DiA are nce i a Time, Tare and Tritt, ez. 16, 17,... 200

ex. 55-64,

.231 Position by I'ractions, ex. 66–76. 240

Barter, ez. N-31,

.234 Discount, ..

....................174
Method of assessing tax :8.ex. 12,13, 215

...202

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