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. 33 Fractions arise from Division, Miscellaneous Questions, involving the Principles of the preceding Rules, 40 Different Denominations, Federal Money, Reduction, COMPOUND NUMBERS. - to find the Value of Articles sold by the 100, or 1000, Tables of Money, Weight, Measure, &c., Addition of Compound Numbers, Subtraction, Multiplication and Division, 44 44 50 53 54 65 66 69 73 FRACTIONS. COMMON or 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,. General Rule for the Multiplication of Fractions, one Fraction by another,. General Rule for the Division of Fractions, Addition and Subtraction of Fractions, Common Denominator, how found, . DECIMAL. Their Notation, Addition and Subtraction of Decimal Fractions, Multiplication of Decimal Fractions, Division of Decimal Fractions, . Reduction of Decimal Fractions, Page To reduce Vulgar to Decimal Fractions, To reduce Shillings, &c., to the Decimal of a Pound, by Inspection, the three first Decimals of a Pound to Shillings, &c., by In- 113 115 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 Same Questions, solved by Analysis, 11 65, ex. 1-20. Having the Diameter of a Circle, to find the Circumference; or, having the Circumference, to find the Diameter, ex. 171-175. ARITHMETIC. NUMERATION. 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, five 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 by the following • ID is used instead of D to represent five hundred, and for every additional annexed at the right hand, the number is increased ten times. CIO is used to represent one thousand, and for every Cand O put at each end, the number is increased ten times. A line over any number intreases fts value one thousand times. Five thousand Fifty thousand Hundred thousand Oue million M. Two million MM A unit, unity, or one, is represented by this character, Two Three Four Five Six Seven Eight Ten has no appropriate character to represent it; but is consi- One ten and one unit are called One ten and seven units are called Two tens are called Three tens are called 6 7 8 9 10 Ten Eleven 11 Twelve 12 Thirteen 13 Fourteen 14 Fifteen 15 Sixteen 16 Seventeen 17 Nine tens are called Ninety 90 Ten tens are called a hundred, which forms a unit of a still Four tens are called higher order, consisting of hundreds, represented by the One hundred, one ten, and one unit, are called One hundred 100 One hundred and eleven 111 2. There are three hundred sixty-five days in a year. In this number are contained all the orders now described, viz. units, tens, and hundreds. Let it be recollected, units occupy the first place on the right hand; tens the second place from the right hand; hundreds the third place. This number may now be decomposed, that is, separated into parts, exhibiting each order by itself, as follows:-The highest order, or hundreds, are three, represented by this character, 3; but, that it may be made to occupy the third place, counting from the right hand, it must be followed by two ciphers, thus, 300, (three hundred.) The next lower order, or tens, are six, (six tens are sixty,) represented by this character, 6; but, that it may occupy the second |