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. Mont Vernon, (N. H.) Sept. 29, 1827. DANIEL ADAMS. Fractions arise from Division, Miscellaneous Questions, involving the Principles of the preceding Rules, Different Denominations, Federal Money, COMPOUND NUMBERS. to find the Value of Arucles sold by the 100, or 1000, Tables of Money, Weight, Measure, &c. Reduction, Addition of Compound Numbers, Subtraction, Multiplication and Division, FRACTIONS. COMMON, OF VULGAR. Their Notation, Proper, Improper, &c. To change an Improper Fraction to a Whole or Mixed Number, a Mixed Number to an Improper Fraction, To reduce a Fraction to its lowest Terms, 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, a Whole Number by a Fraction, one Fraction by another, General Rule for the Multiplication of Fractions, General Rule for the Division of Fractions, Common Denominator, how found, 101 102 103 104 105 196 107 110 112 113 114 115 117 118 119 120 121 Rule for the Addition and Subtraction of Fractions, 124 Reducuon 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 Shilings, &c., by Inspection, 157 Reduction of Currencies, To reduce English, &c. Currencies to Federal Money, Interest, To find the Interest on Notes, Bonds, &c., when partial Payments have Compound Interest, 153 154 155 156 164 165 166 167 168 169 by Progression, 229 Equation of Payments, 176 Ratio, or the Relation of Numbers, Proportion, or Single Rule of Three, 179 Same Questions, solved by Analysis, ¶ 65, ex. -20. Compound Proportion, or Double Rule of Three, 187 Fellowship, Taxes, Method of assessing, Alligation, 197 Duodecimals, 201 Involution, Extraction of the Square Root, Application and Use of the Buying and Selling Stocks, ¶ 82.7 Square Root, see Supplement, Cube Root, see Supplement, Commission, T 82; 11 85,ex. 5, 6. Scale for taking Dimensions in Feet and Decimals of a Foot, 204 207 207 212 215 220 225 237 MISCELLANEOUS EXAMPLES. Barter, ex. 21-32. Position, ex. 89-108. 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 To find the Area of a Circle, ex. 176-179. of a Globe, ex. 180, 181. NUMERATION. 1. 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, five hundred, and M, one thousand. As often as any letter is repeated, so many times its value is repeated, unless it De 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, ning &c., as will be seen in the following *I is used instead of D to represent five hundred, and for every additional nexed at the right hand, the number is increased ten times. an † CIO is used to represent one thousand, and for every C and Ɔ put at each end, the number is increased ten times. A line over any number increases its value one thousand times. A unit, unity, or one, is represented by this character, 1 2, 3 4. 5. 6. 7. 8. 9. Ten has no appropriate character to represent it; but is Ten 10 11. One ten and two units are called One ten and three units are called Two tens are called Three tens are called Four tens are called Five tens are called Eleven Nine tens are called Ninety Ten tens are called a hundred, which forms a unit of a still higher order, consisting of hundreds, represented by the same character (1) as a unit of each of the foregoing orders, but is written one place further toward the left hand, that is, on the left hand side of tens; thus, One hundred 100. One hundred, one en, and one unit, are called |