ling; then call the figures in the second and third places, after 5, if contained, is deducted, so many farthings; abating 1, when they are above twelve; and 2, when above 36; and the result is the answer. EXAMPLES. 1. Find the value of '7851. by inspection. 1s. for 5 in the place of tenths. for the excess of 12, abated. 15s. 8d. the answer. 2. Find the value of 8751. by inspection. Ans. 17s. 6d. 3. Value the following decimals by inspection, and find their sum, viz. 9271. +3511. +2031. +0611.021. + 10091. Ans. 11. 11s. 51d. FEDERAL MONEY.* THE denominations of Federal Money, as determined by an Act of Congress, Aug. 8, 1786, are in a decimal ratio; and therefore may be properly introduced in this place. * The coins of federal money are two of gold, four of silver, and two of copper. The gold coins are called an eagle and halfeagle; the silver, a dollar, half-dollar, double dime, and dime; and the copper, a cent and half-cent, The standard for gold and silver is eleven parts fine and one part alloy. The weight of fine gold in the eagle is 246'268 grains; of fine silver in the dollar, 375 64 grains; of copper in 100 cents, 21lb. avoirdupois. The fine gold in the half-eagle is half the weight of that in the eagle; the fine silver in the half-dollar, half the weight of that in the dollar, &c. The denominations less than a dollar are expres sive of their values: thus, mill is an abbreviation of mille, a thousand, for 1000 mills are equal to 1 dollar; cent, of centum, a hun A mill, which is the lowest money of account, is .001 of a dollar, which is the money unit. A cent A dime A dollar An eagle is '01 Or 10 mills 1 cent. 10 dollars eagle, = eagle, E. A number of dollars, as 754, may be read 754 dollars, or 75 eagles, 4 dollars; and decimal parts of a dollar, as 365, dred, for 100 cents are equal to 1 dollar; a dime is the French of tithe, the tenth part, for 10 dimes are equal to 1 dollar. The mint-price of uncoined gold, 11 parts being fine and 1 part alloy, is 209 dollars, 7 dimes, and 7 cents per lb. Troy weight; and the mint-price of uncoined silver, 11 parts being fine and 1 part alloy, is 9 dollars, 9 dimes, and 2 cents, per lb. Troy. In Mr. PIKE'S "Complete System of Arithmetic," may be seen "RULES for reducing the Federal Coin, and the Currencies of the several United States; also English, Irish, Canada, Nova-Scotia, Livres Tournois, and Spanish milled dollars, each to the par of all the others." It may be sufficient here to observe respecting the currencies of the several States, that a dollar is equal to 6s. in New-England and Virginia; 8s. in New-York and North-Carolina; 7s. 6d. in New-Jersey, Pennsylvania, Delaware, and Maryland; and 4s. 8d. in South-Carolina and Georgia. The English standard for gold is 22 carats of fine gold, and 2 carats of copper, which is the same as 11 parts fine and 1 part alloy. The English standard for silver is 18oz. 2dwt. of fine silver, and 18dwt. of copper; so that the proportion of alloy in their silver is less than in their gold. When either gold or silver is finer or coarser than standard, the variation from standard is estimated by carats and grains of a carat in gold, and by penny-weights in silver. Alloy is used in gold and silver to harden them. NOTE. Carat is not any certain weight or quantity, but of any weight or quantity; and the minters and goldsmiths divide it into 4 equal parts, called grains of a carat. may be read 3 dimes, 6 cents, 5 mills, or 36 cents, 5 mills, or 365 mills; and others in a similar manner. Addition, subtraction, multiplication, and division of federal money are performed just as in decimal fractions; and consequently with more ease than in any other kind of money. EXAMPLES. 1. Add 2 dollars, 4 dimes, 6 cents, 4D. 2d., 4d. 9c., 1E. 3D. 5c. 7m., 3c. 9m., 1D. 2d. 8c. 1m., and 2E. 4D. 7d. 7. Multiply 3D. 4d. 5c. 1m, by 1D. 2d. 3c. 2m. IT has already been observed, that when an infinite decimal repeats always one figure, it is a single repetend; and when more than one, a compound repetend; also that a point is set over a single repetend, and a point over the first and last igures of a compound repetend. It may be farther observed, that when other decimal figures precede a repetend in any number, it is called a mixed repetend: as '23, or 104123; otherwise it is a pure, or simple, repetend: as 3 and 123. Similar repetends begin at the same place: as 3 and 6, or 1:341 and 2'156. Dissimilar repetends begin at different places: as 253 and '4752. Conterminous repetends end at the same place: as 125 and *009. Similar and conterminous repetends begin and end at the same place: as 2'9104 and '0613. REDUCTION OF CIRCULATING DECIMALS. CASE I. To reduce a simple repetend to its equivalent vulgar fraction. RULE*. 1. Make the given decimal the numerator, and let the denominator be a number, consisting of as many nines as there are recurring places in the repetend. 2. If there be integral figures in the circulate, as many cyphers must be annexed to the numerator, as the highest place of the repetend is distant from the decimal point. * If unity, with cyphers annexed, be divided by 9 ad infinitum, the quotient will be 1 continually; i. e. if be reduced to a deci mal, it will produce the circulate 1; and since 1 is the decimal i equivalent to, 2 will, 33, and so on till '9==1. Therefore every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and denominator 9. Again, or, being reduced to decimals, makes '010101, &c. or '001001, &c. ad infinitum '01 or 001; that is, 01, and001; consequently 02, 03, &c. and = 002,003, &c. and the same will hold universally. |