foregoing, and the sum of these three numbers is called the Jubtrahend. 7. Then subtract this subtrahend from the resolvend, and to the remainder bring down the next period for a new resolvend, with which proceed as before, and so on till the work be finished. Example 4. What is the cube root of 164566592 ? 164566592(548 cube of the first period 765) 39566 first refolvend 15 three times the root 5 04 cube of 4 32404 fubtrabend 86742) 7502592 second resolvend 162 three times 54 512 cube of 8 Answer 548 the cube root requirecta There are many other methods of extracting the cube ront given by different mathematicians; but none of which I have seen are so simple and easy to be remembered as the method here laid down. After 'what has been said, it need hardly be mentioned, that the extraction of the square root is proved by multiplying the root into itself; and the extraction of the cube root, by multiplying the root three times into itself. Kk 2 СНАР. CHAP. IV. OF DECIMAL ARITHMETIC, . SECT. I. REDUCTION OF DECIMAL FRACTIONS. A DECIMAL fraction is that fraction whose denominator has an unit in the first place on the left hand, with as many cyphers annexed as necessary : thus to do Trbios &c. are decimal fractions. But these fractions are usually expressed in writing without a denominator, by writing the numerator, and prefixing as many points or cyphers before.. it on the left hand, as there are more places of figures in the denominator than in the numerator. Thus the foregoing fractions are written .5, .25, .048, .0572, and expressed five tenths, eventy-five hundredths, forty-eight thousandths, &c. Cyphers placed on the right hand of decimal fractions make no alteration in their value: thus, if a cypher be annexed to the foregoing fraction i5, it will be then .50 fifty hundredths or half an integer, as before ; if two cyphers .500 it will be five hundred thousandths, or i, and the fame of the others. But qyphers placed on the left band of a decimal fraction decrease their value, every cypher decreasing it in a tenfold proportion: thus .5, .05, .005, are five tenths, five hundredths, fave thousandths parts respectively. The The value of every figure in a decimal fraction increases in a tenfold proportion from the first place on the right hand, as in whole numbers. The figures in the first place of a decimal fraction on the left hand are called primes, those in the second place seconds, those in the third place thirds, &c. DECIMAL TABLES OF Coin, Weight, and Measure. 2 2 3004166 1 11 18,9 2 2 2016666 6 10125 TABLE I. / TABLE II. 3 10155 $ 1,010416 -008333 4 ,008333 English Coin. English Coin 1s. 1,0041666 3 ,00625 11. and is the In-Long Meas. 1 foot teger. Troy Weight 1oz. Grains. TDecimais. 2002083 Sh. Dec. Sk. Doc. Inches. Decimals, ,001910 191,95 | 91,45 6 ,5 10 1,001736 8,4 TABLE IV. 5 2416666 9 3001562 171,857,35 ,333333 8 ,001389 Avoir dup. Wt. 16,8 61,3 ,25 7 ,001215 15] 275 5,25 ,166666 6 9001042 1121b. the Integer 141 27 ,083333 5 ,c00868 Qrs. Decimals. 13.65 131,15 4 Farths. Decimals. ,000694 ,75 12,6 3 ,9005:1 - 95 111,55 1,05 3 ,0625 ,000347 ,25 101 5 1041666 1,020833 ,000173 Pounds. Decimals, Pence. Decimals ,125 13 1116071 ,107143 9016666 Troy Weight. Pennyweights the II 4 ,098214 3 ,0125 ,089286 ilb. the Integer, fome as Shillings 9 in the Fir/Table. ,080357 2004166 8. 1071428 weight. Decimals. Grains. Decimals 7 ,0625 10 ,041666 1,2 1,025 6 Farths. Decimals. ,053571 20375 11 ,022916 5 ,044643 3 ,003125 8 ,033333 1,020833 4 ,035714 ,0020833 7 9029166 9' 018757 3,026786 1,0010416 6 ,025 ,097857 5 ,020833 7 1,014583 ,008928 2 I 2 14 I 2 10 ,008333 Penny 1 1 2 2 1 2 I 1 3 2 2 Ounces. Decimals. Drams. Decimals. 7 ,027777 i Pints. Decimals. 8,004464 8,03125 6 ,023809 3 ,005952 7 ,027343 5 ,019841 1003968 ,015873 ,001984 4002790 5 9019531 3. 011904 4 ,902232 4 20.15525 ,007936 TABLE VII. 1,001674 3 1,011718 ,003968 ,001116 ,007812 Pints. I'Decimals. Liquid. Drg: Menfare. 1,000558 ,003906 4 ,001984 Gall. i Quarter Joz. Decimals. 3 ,001488 the Integer. 3 9000418 ,000992 Pinssi Dee. Baf. ,000279 | TABLE VI. ,000446 4 ,5 4 ,000139 Liquid Meafure ,,395 Hogmead the ,25 TABLE V. u Tun the Integer. Integer. ,125 GallonsDecimals Gallons Decimals. Avoir dup. W. 100 4 pts. Dec. IPA ,396825 30 1476190 1,0937513 ilb. the Integer. 90 ,357141 0317460 ,0625 80 1317460 1,158730 Ounces. Decimals. ,0312511 227772 9 ,142857 18 ,5 Decimals.14 pecks 60 ,238095 ,126984 94375 50 ,198412 3 40 ,158730 6 5 :,3125 ,119047 5 + -5 20 ,079365 4 ,063492 Decimals. Pints 3 ,1875 ,039682 3 ,641619,005859 ,125 7035714 1,031745,003906 1,0625 ,031746 1,0158731,001953 1 20 2 I 70. 1. 'Caser: To reduce a vulgar fraction to a decimal one of the same value. Rules Divide the numerator by the denominator, and the quotient will be the decimal fraction required. If the numerator be too small to be divided by the denominator, add one or more cyphers thereto, in which case as many places must be separated from the quotient for decimals as the difference of the number of places of 'decimals in the dividend and the divisor, and the remaining figures in the quotients (if any) are integers. But if the number of places in the quotient be less than the required number of decimal places, as many cyphers must be prefixed on the left hand thereuf as are necesary. Example 1. What is the decimal fraction of a pound equal to ? Here o Here two cyphers must be added to the nume: 4) 300 rator, which divided by 4 quotes 75, which I 75 mark for decimals, as there are two decimals added to the dividend, and the qaotient is the seventy-five hundredth part of a pound, or 2, equal to 158 ... Example 2. What is the decimal fraction of a pound equal to 9d.? - Ans. .0375. As there are but three places of figures 24,0)900601.0375 in the quotient, I place a cypher on the left thereof, for there should be four figures pointed off for decimals, being so many cypliers annexed to 97' thus the quotient is 13363 of a pound. Example 3: What is the decimal fraction of a pound for 3 farthings ? - Ans. .003125. · Here the vulgar fraction for 3 96,0) 300000,00.003125 farthings is 7o; I therefore place as many cyphers as are necessary to the numerator 3, which in this case is 6 cyphers, and the quotient is 3125, but there must be 6 decimals in the quotient, as there are 6 in the dividend and none in the divisor; I therefore prefix two cyphers to the left hand thereof as before directed. Example 4. . What is the decimal fraction of a year for 73 days ? - Answer.2, equal to nos 365)7300.2 Cafe 2. To find the value of a decimal fraction in money, weight, or measure. Rule. Multiply the decimal fraction by the number of parts of the next inferior denoinination, and from the produet cut off as many figures from the right hand side as there are decimal places in the fraction. Then multiply these figures fo separated on the right hand by the number of parts of the next inferior denomination, and from the product cut off as many figures as this last multiplicand has decimal places, as befere. Proceed in the same manner through all the denominations, then the separated figures on the left hand side will be the answer. Example |