AZ THERE are two methods of teaching,—the synthetic and the analytic. 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 ele. mentary 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, by 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 pot tronage, 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 ap. plication. 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 solution 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 member 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. DANIEL ADAMS. Mont Vernon, (N. H.) Sept. 29, 1827. INDEX Page 7 Fractions arise from Division, Miscellaneous Questions, involving the Principles of the preceding Rules, 52 to find the value of Articles sold by the 100, or 1000, . Common, or VULGAR.. Their Notation, To change an Improper Fraction to a Whole or Mixed Number, 105 Greatest common Divisor, how found, 106 To divide a Fraction by a Whole Number; two ways, 107 To multiply a Fraction by a Whole Number; two ways, 110 a Whole Number by a Fraction, 112 one Fraction by another, 113 General Rule for the Multiplication of Fractions, 114 To divide a Whole Number by a Fraction, 115 one Fraction by another, . 117 General Rule for the Division of Fractions, 118 Addition and Subtraction of Fractions, 119 Common Denominator, how found, 120 Least Common Multiple, how found, 121 Rule for the Addition and Subtraction of Fractions, 124 Reduction of Fractions, 124 DECIMAL. Their Notation, 132 Addition and Subtraction of Decimal Fractions, 135 Multiplication of Decimal Fractions, 137 Division of Decimal Fractions, 139 To reduce Vulgar to Decimal Fractions, 142 Reduction of Decimal Fractions, 145 To reduce Shillings, &c., to the Decimal of a Pound, by Inspection, 146 the three first Decimals of a Pound to Shillings, &c., by Inspection, 157, . Page To reduce English, &c. Currencies to Federal Money, Federal Money to the Currencies of England, &c. one Currency to the Par of another Currency, 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., and Interest given, to find the Time, To find the Interest on Notes, Bonds, &c., when partial Payments have Rátio, or the Relation of Numbers, Proportion, or Single Rule of Thrée, Same Questions, solved by Analysis, 1 65, ex. 1—20. Compound Proportion, or Double Rule of Three, Taxes, Method of assessing, 195 Scale for taking Dimensions in Feet and Decimal of a Foot, 204 Extraction of the Square Root, Application and Use of the Square Root, see Supplement, Application and Use of the Cube Root, see Supplement, Arithmetical Progression, 222 Geometrical Progression, Annuities at Compound Interest, 231 Permutation, Practice, fi 29, ex. 10–19. || 43. Commission, T 82; 1.85, ex. 5, 6. Barter, ex. 21–32. | Position, ex. 89—108. To find the Area of a Square or Parallelogram, ex. 148–154. Hlaving the Diameter of a Circle, to find the Circumference; or, having the Circumference, to find the Diameter, ex. 171-175. To find the Area of a Circle, ex. 176—179. To find the Solid Contents of a Globe, ex. 182—184. of a Pyranid, or Cone, ex. 188, 189. of any Irregular Body, ex. 202, 203. | Mechanical Powers, ex. 192—201. 1 NUMERATION, 1 1. A SINGLE or individụal 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 TABLE. LXXXX. or XC. Two il, One hundred C. Two hundred CC. lIIl. or IV. Three hundred CCC. Four hundred CCCC. D. or 10.* Six hundred DC. Seven hundred DCC. Eight hundred DCCC. Nine hundred DCCCC. One thousand M. or CIO. Five thousand 155. or. XXXX. or XL. Ten thousand CC103.or X. Fifty thousand 1203. Hundred thousand CCCIO39. or T. One million M. Two million MM. * 15 is used instead of D to represent five hundred, and for every additional ar dexed at the right hand, the number is increased ten times. † CI5 is used to represent one thousand, and for every C and I put at each end, the number is increased ten times. | A line over any number increases its value one thousand times. |