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

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 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. ls 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 probable 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 comion 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 alinost 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 tan either. But, 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 harmoniqus relations of numbers. Here is a delightful field for ar inquisitive mind. It here imbibes truths as lasting as life. When the utility and convenience of this rule are once conceded, ail 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 atten: tion is most needed on other subjects. With as much propriety the name on 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

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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 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 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 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 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 ra tionally, 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 almost every learner, as to reserve this interesting portion of mathematics for a favoured few, and, in the mean time, to divert the atten lion 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, Mensuraton, &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 iess 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 fac-, tors are each half an inch, but the solid contents are much less than half a solid inch.

In this work, the author has endeavoured to make every part conform to this maxim, víz. 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 comp hension 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 he 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; 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 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 Destiff, 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 intelligence and good sense. It is easy to make a boy, who does not reason, Tepeat, by rote, any technical rules, which a common writing master, with magisterial solemnity, may lay down for him ; but a child who reasons will not be thas easily managed'; he stops, frowns, hesitates, questions his master, is wrelched 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-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 kno" 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 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 mcasuring all things that are capable of augmentation or diminution ; and, when rationally tanght, 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

THE AUTHOR

SIMPLE NUMBERS.

.

COMPOUND NUMBERS.

Federal Money-different Denominations,

Reduction of Federal Money,

Addition of Federal Money,

Subtraction of Federal Money,

Multiplication of Federal Money,

Division of Federal Money,

To multiply

by 1, 3, 5}, &c.,

Practice in Federal Money,

Questions on the foregoing-Bills of goods sold,

Reduction-Tables of Money, Weight, Measure, &

Compound Addition,

Compound Subtraetion,

Coinpound Multiplication,

Compound Division,

Questions on the foregoing,

33

56

57

58

60

61

03

64

C6

69

89

93

97

98

100

FRACTIONS.

Conmon, or Vulgar, :
Frations arise from Division,
Proper, Improper, &c.,

an Improper Fraction to a Whole or Mixed Namber,

To change a Mixed Number to an Improper Fraction,

To reduce a Fraction to its lowost Terms,

To change

101

103

104

104

105

107

.

Pagn.

To multiply a Fraction by a Whole Number,

108

To multiply a Whole Numhver by a Fraction,

110

To multiply one Fraction by another,

112

To find the Least Common Multiple of two or more Numbers,

114

To find the Greatest common Divisor of two or more Numbers, (reference,) 115

To reduce Fractions of Different Denominators to a Common Denominator, 117

Adilition of Fractions,

119

Subtraction of Fractions,

121

Division of Fractions-To divide a Fraction by a Whole Number, 122

To divide a Whole Number by a Fraction,

To divide one Fraction by another,

124

To reduce Whole Numbers to the fraction of a greater Denomination, 125

To reduce a Fraction to Whole Numbers of less Denominations,

126

To reduce Fractions of a higher Denomination to lower,

To reduce Fractions of a lower Denomination into a higher,

129

DECIMALS.—Their Table of Notation,

130

Addition of Decimals,

133

Subtraction of Decimals,

134

Multiplication of Decima.s,

135

Division of Decimals,

137

To change Vulgar or Common Fractions to Decimals,

140

Circulating Decimals, (reference,)

141

To reduce Compound Numbers to Decimals of the highest Denomination, 142

To reduce Decimals of higher Denominations to Whole Numbers of lower

Denominations,

144

Reduction of Currencies,

145

Questions on the foregoing,

152

Interest,

155

Concise "Rule for calculating Interest in New York State,

161

Commission, Insurance, Stock, Loss and Gain,

162_3

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

164

Time, Raie per cent., and Interest being given, to find the Principal, 166

The Principal, Interest, and Time, being given, to find the Rate per cent., 166

The Principal, Rate per cent., and Interest, being given, to find the Time, 16e

Compound Interest-Compound Interest by Table,

108

Equation of Payments,

172

Questions on the foregoing,

173

Rule of Three, by Analysis,

174

Ratio, or the Relation of Numbers,

177

Proportion arising from Ratio, or from Multiplication and Division,

178

Application of Ratio by Rule,

181

Rule of Threo in Vulgar and Decimal Fractions,

185

Compound Proportion, or Double Rule of Three, by Ratio and Analysis, 187

Questions on the foregoing,

To compute the Interest on Notes with Endorsements--three modes, : 195

Practice in Compound Numbers,

200

Fellowship-by Analysis—by Ratio,

201

Compound Fellowship-by Analysis_by Ratio,

204

Mensuration Square Measure,

206

209

Solid, or Cubic Measure,

212-13

Duodecimals—Multiplication of Duodecimals,

Questions on the foregoing,

215

To calculate Difference in Time, Tare and Trett, ez. 16, 17, 194

ex. 55-64, .

217 Position by Fractions, ex. 66—76, 26

164

Barter, ex. 2–31,

220 | Discount,
Method of assessing taxes,ez.12,13, 204

190

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