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be written. In the beginning, a single order, only, is employed; afterwards, 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 con. trived, 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 cannut 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 ho is required to remember without comprehending, and tired of the irk. some 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 con. tinue 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 con. tents, 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 decimals, 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 con. formity 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 de. mands. 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 singlo 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.

io. 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 af. ter 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 ob. jects, 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 cur 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 oź these par. ticulars has any deviation been made from other writers, without long and patient observation of the effect of different methods of instruc. tion upon the youthful mind. This is the test to which the author has subjected all his proposed improvements, and the uniform success wbich has attended his experiments, leads him to look with confidence for the approbation of the public.

HartFORD, JULY 30, 1830.

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Origin of measures of length, of surface, of capacity and

135
of solidity,

Origin of weights, and the balance,

136

Denominations of coin,

Division of time,

Calculations in simple and compound numbers compared,

THE FRENCH MONEY, WEIGHTS AND MEASURES,

137

British do.

139

New York & Connecticut do.

140

Promiscuous Examples,

RULE OF THREE by Analysis,

142

FRACTIONS,-general principles,

To reduce them to their lowest terms,

144

Prime numbers--measures—to determine the measures

145

of a number,

To find cominon measures-a greatest com. meas.

146

To find a com. denominator,

148

Multiples--a least com. mult.,

149

To find a least com, den.,

151

Addition,

153

Multiplication,-a Fraction by a whole number,

154

A whole number by a Fraction,

156

Practice,

To multiply a Fraction by a Fraction,

157

Compound Fractions,

158

FELLOWSHIP,-simple and compound, by analysis and ratio, 159-61

Subtraction,

162

Division,-a Fraction by a Fraction,

163

A whole number by a Fraction,

164

A Fraction by a whole number,

166

To reduce whole numbers to Fractions of higher denominations, 167

fractions to do.

168

fractions to do. of lower denom.

169

fractions to whole nnmbers of lower densm.

170

DECIMALS,-general principles,

To reduce Vulgar Fractions to Decimals,

174

REPETENDS, OR CIKCULATING DECIMALS,

175 & 78

To reduce whole numbers to decimals of higher denominations, 176

Approximates,

Contraction for Sterling Money,

177

Addition,

178

Multiplication,

179

Contraction in Multiplication,

181

To reduce decimals to whole numbers, of lower denom., 182

Contraction for Sterling Money,

183

Subtraction,

Division, .

184

Contraction in Division,

186

Another mode of reducing whole numbers to decimals of

187

higher denom.,
Theory of circulates,

188

Arithmetical operations upon them,

193

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