besides, will serve to interest him in the science, since he will find himself alle, by the application of a very few principles, to solve many curious questions. The arrangement of the subjects is that which to the author bas 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, s relained, 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 hy classes ought to be instituted in every school where arithmetic is taught. The supplements io 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. SIMPLE NUMBERS. Pago Numeration and Notation, 11 16 22 29 33 COMPOUND NUMBERS. 44 53 54 65 66 69 73 44 79 84 FRACTIONS. a Mixed Number to an Improper Fraction, Greatest Common Divisor, how found, a Whole Number by a Fraction, one Fraction by another, one Fraction by another, Common Denominator, how found, - Least Common Multiple, how found, 89 116 134 Pago To reduce Vulgar to Decimal Fractions, 113 Reduction of 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- 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., Principal, Rate per cent., and Interest given, lo find the Time, To find the Interest on Notes, Bonds, &c., when partial Payments have Ratio, or the Relation of Nanibers, Proportion, or Single Rule of Three, Same Questions, solved by Analysis, 1 65, ex. 1-20. Compound Proportion, or Double Rule of Three, Scale for taking Dimensions in Feet and Decimals of a Foot, 156 Extraction of the Square Root, Application and Use of the Square Root, see Supplement, Application and Use of the Cube Root, see Supplement, 165 | Geometrical Progression, Annuities at Compound Interest, 169 | Permutation, Practice, 1 29, ex. 10–19. 43. Commission, 1 82; T 85, ex. 5, 6. To find the Area of a Square or Parallelogram, ex. 148-154. Having the Diameter of a Circle, to find the Circumference; or, having tho 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 Pyramid, or Cone, ex. 188, 189, of any Irregular Body, ex. 202, 203, | Mechanical Powers, ex. 192–201. 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 seren; seven and one more are called eighi ; cight 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, fire; 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, un. „ess it be a letler representing a less number placed before one representing a greater; then the less number is taken from the greater ; thus Iû represents four, IX nine, &c., as will be seen by the following TABLE. LXXXX.or XC One hundred C. Two hundred CC. IIII. or IV. Three hundred CCC. Four hundred CCCC. Five hundred D. or 13.* Six hundred DC. Seven hundred DCC. VIIII. or IX. Eight hundred DCCC. Nine hundred DCCCC. One thousand M. or CIO. Thirty XXX. Five thousand 155. or 7.1 Forty XXXX.01 XL. Ten thousand CCIDO. or X. L. Fifty thousand 1503. Sixty LX. Hundred thousand CCCI.or 7. Seventy LXX. Oue million N. Eighty LXXX. Two million MM. • 1is used instead of D to represent five hundred, and for every additiona: 3 annexed at the right hand, the number is increased ten times. Cl3 is used to represent one thousand, and for every C and put aleach önd, the number is increaned ten times. * A lme bver any number increasew to value one thousand timtes. Fifty 1 2 3 4 5 6 7 8 9 11 "A unit, unity, or one, is represented by this character, dered as forming a unit of a second or higher order, consist- Ten 10 One ten and one unit are called Eleven One ten and two units are called Twelve 12 One ten and three units are called Thirteen 13 One ten and four units are called Fourteen 14 One ten and five units are called Fifteen 15 One ten and six units are called Sixteen 16 One ten and seven units are called Seventeen 17 One ten and eight units are called Eighteen 18 One ten and nine units are called Nineteen 19 Two tens are called Twenty 20 Three tens are called Thirty 30 Four tens are called Forty 40 Five tens are called Fifty 50 Six tens are called Sixty 60 Seven tens are called Serenty 70 Eight tens are called Eighty Nine tens are called Ninety, 90 Ten tens are called a hundred, which forms a unit of a still higher order, consisting of hundreds, represented by the is, on the left hand side of tens ; thus, One hundred 100 One hundred, one ten, and one unit, are called One hundred and eleven 111 12. 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 8C |