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THERE are two methods of teaching, the synthetic and the analyti In the synthetic method, the pupil is first presented with a general view of the science he is studying, and afterwards with the particulars of which it consists. The analytic method reverses this order: the pupil is first presented with the particulars, from which he is led, by certain natural and easy gradations, to those views which are more general and comprehensive.
The Scholar's Arithmetic, published in 1801, is synthetic. If that is a fault of the work, it is a fault of the times in which it appeared. The analytic or inductive method of teaching, as now applied to clementary instruction, is among the improvements of later years. Its introduction is ascribed to PESTALOZZI, a distinguished teacher in Switzerland. It has been applied to arithmetic, with great ingenuity,
Mr. COLBURN, in our own country.
The analytic is unquestionably the best method of acquiring know ledge; the synthetic is the best method of recapitulating, or reviewing it. In a treatise designed for school education, both methods are useful. Such is the plan of the present undertaking, which the author, occupied as he is with other objects and pursuits, would willingly have forborne, but that, the demand for the Scholar's Arithmetic still continuing, an obligation, incurred by long-continued and extended patronage, did not allow him to decline the labour of a revisal, which should adapt it to the present more enlightened views of teaching this science in our schools. In doing this, however, it has been necessary to make it a new work.
In the execution of this design, an analysis of each rule is first given, containing a familiar explanation of its various principles; after which follows a synthesis of these principles, with questions in form of a supplement. Nothing is taught dogmatically; no technical term is used till it has first been defined, nor any principle inculcated without a previous developement of its truth; and the pupil is made to understand the reason of each process as he proceeds.
The examples under each rule are mostly of a practical nature, beginning with those that are very easy, and gradually advancing to those more difficult, till one is introduced containing larger numbers, and which is not easily solved in the mind; then, in a plain, familiar manner, the pupil is shown how the solution may be facilitated by figures. In this way he is made to see at once their use and their application.
At the close of the fundamental rules, it has been thought advisable to collect into one clear view the distinguishing properties of those rules, and to give a number of examples involving one or more of them These exercises will prepare the pupil more readily to understand the
application of these to the succeeding rules; and, besides, will serve to interest him in the science, since he will find himself able, by the application of a very few principles, to solve many curious questions.
The arrangement of the subjects is that, which to the author has appeared most natural, and may be seen by the Index. Fractions have received all that consideration which their importance demands. The principles of a rule called Practice are exhibited, but its detail of cases is omitted, as unnecessary since the adoption and general use of federal money. The Rule of Three, or Proportion, is retained, and the solu tion of questions involving the principles of proportion, by analysis, is distinctly shown.
The articles Alligation, Arithmetical and Geometrical Progression, Annuities and Permutation, were prepared by Mr. IRA YOUNG, a mem ber of Dartmouth College, from whose knowledge of the subject, and experience in teaching, I have derived important aid in other parts of the work.
The numerical paragraphs are chiefly for the purpose of reference these references the pupil should not be allowed to neglect. His attention also ought to be particularly directed, by his instructer, to the illustration of each particular principle, from which general rules are deduced for this purpose, recitations by classes ought to be instituted in every school where arithmetic is taught.
The supplements to the rules, and the geometrical demonstrations of the extraction of the square and cube roots, are the only traits of the old work preserved in the new.
Mont Vernon, (N. H.) Sept. 29, 1827.
Numeration and Notation,
Fractions arise from Division,
Miscellaneous Questions, involving the Principles of the preceding Rules,
to find the Value of Articles sold by the 100, or 1000,
2 Bills of Goods sold,
Tables of Money, Weight, Measure, &c.
COMMON, OF VULGAR. Their Notation,
To change an Improper Fraction to a Whole or Mixed Number,
a Mixed Number to an Improper Fraction,
Greatest common Divisor, how found,
To divide a Fraction by a Whole Number; two ways,
To multiply a Fraction by a Whole Number; two ways,
General Rule for the Multiplication of Fractions,
To divide a Whole Number by a Fraction,
General Rule for the Division of Fractions,
Common Denominator, how found,
Least Common Multiple, how found,
DECIMAL. Their Notation,
Addition and Subtraction of Decimal Fractions,
Multiplication of Decimal Fractions,
Division of Decimal Fractions;
To reduce Vulgar to Decimal Fractions,
Reduction of Decimal Fractions,
To reduce Shillings, &c., to the Decimal of a Pound, by Inspection,
To reduce English, &c. Currencies to Federal Money,
Time, Rate per cent., and Amount given, to find the Principal,
Time, Rate per cent., and Interest given, to find the Principal,
Principal, Interest, and Time given, to find the Rate per cent.,
Principal, Rate per cent., and Interest given, to find the Time,
To find the Interest on Notes, Bonds, &c., when partial Payments have
Ratio, or the Relation of Numbers,
Proportion, or Single Rule of Three,
Same Questions, solved by Analysis, ¶ 65, ex. 1-20.
Compound Proportion, or Double Rule of Three,
Extraction of the Square Root,
Application and Use of the Square Root, see Supplement,
Annuities at Compound Interest,
Practice, 29, ex. 10-19. ¶ 43.
of a Globe, ex. 180, 181.
Gauging, ex. 190, 191.
Forms of Notes, Bonds, Receipts, and Orders,
of a Pyramid, or Cone, ex. 188, 189.
Commission, T 82; ¶ 85, ex. 5, 6.
Barter, ex. 21-32.
To find the Area of a Square or Parallelogram, ex. 148–154
Having the Diameter of a Circle, to find the Circumference; or, having the
Circumference, to find the Diameter, ex. 171–175.
11. A SINGLE or individual thing is called a unit, unity, or one; one and one more are called two; two and one more are called three; three and one more are called four; four and one more are called five; five and one more are called six; six and one more are called seven; seven and one more are called eight; eight and one more are called nine; nine and one more are called ten, &c.
These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words; one called the Roman method by letters,* and the other the Arabic method by figures. The latter is that in general use.
In the Arabic method, the nine first numbers have each an appropriate character to represent them. Thus,
* In the Roman method by letters, I represents one; V, five; X, ten; L, fifty: C, one hundred; D, fire hundred; and M, one thousand.
As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater; then the less number is taken from the greater; thus, IV represents four, IX, nine, &c., as will be seen in the following
Lo is used instead of D to represent five hundred, and for every additional ♫ an ́ exed at the right hand, the number is increased ten times.
+ CIO is used to represent one thousand, and for every C and Ɔ put at each end, tâm
samber is increased ten times.
A time over any number increases its value one thousand times, .