MULTIPLICATION. RULE.—1. Write the multiplicand, and under it the multiplier, in the same manner as simple numbers; then multiply without regard to decimal points. 2. When the multiplication is finished, begin at the right hand figure of the product and count off as many figures toward the left as there are decimal places in the multiplier and multiplicand, and there place the decimal point. 3. If the number of places in the product be less than the decimal places in the multiplier and multiplicand, prefix a sufficient number of ciphers, to the left of the product, to equal those of the multiplier and multiplicand, and then place the decimal point to the left of the ciphers. DIVISION. RULE.—Divide as in whole numbers, and point off as many figures for decimals in the quotient as the decimal places in the dividend exceed those in the divisor. If the quotient does not contain figures enough, supply the deficiency by prefixing ciphers. To reduce a decimal to a common or vulgar fraction. RULE.—Erase the decimal point; then write the decimal denominator under the numerator, and it will form a common fraction, which may be treated in the same manner as other common or vulgar fractions. To reduce a common or vulgar fraction to a decimal. RULE.-Annex ciphers to the numerator, and divide it by the denominator. Point off as many decimal figures in the quotient as you have annexed ciphers to the numerator. To reduce several denominations to the decimal of a higher deno mination. RULE.—1. Multiply by as many as it takes of the next lower denomination to make one of the higher, adding in the denominations respectively as you multiply, until they are reduced to the lowest denomination in the question, and this is the dividend. 2. Then take one of that denomination of which you wish to make it a decimals and reduce it to the same denomination with the one above-mentioned, and this last number is the divisor. 3. Divide as in whole numbers, and the quotient is the anTo reduce a decimal to its proper value, or compound number to swer. whole numbers of lower denominations. RULE.-1. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for a remainder (counting from the right) as there are decimal places in the given decimal, and there make the decimal point. 2. Multiply the remainder (that is, the decimal) by the next less denomination, and cut off à remainder as before; continue in this way through all the parts of the integer, and the several denominations standing on the left of the decimal points is the answer. REVIEW What are fractions ? What are decimal fractions? From what do they arise? Why are they called decimals ? Ans. Because they decrease in à tenfold ratio, as tenths, hundredths, &c. How are decimals expressed ? What is always the denominator of a decimal fraction ? What is the point placed before a decimal called ? Upon what does the value of a decimal depend? What is the difference between prefixing and annexing ciphers to decimals? How are decimals read ? Repeat the process of addition-subtraction-multiplication-division-reduction, &c. QUESTIONS 1. Add 12-34565, 7-891, 2-34, 14, 0011 together. = 36.77 $ 2. Add 7509, .0074, 69, 8408, 6109 together. 3. Add 7569, 25, 654, 199 together. 4. Add 71.467, 27.94, 16.084, 98.009, 86.5 together. 5. Add 9607.84, 823.79, 07965, 74.821 together. 6. Add 19.073, 2:3597, 223, 0197581, 3478.1, 12:358 together. 7. Add 5-3, 11-973, 49, 9, 1.7314, 34-3 together. 8. From 125.61000 take 95.58756. 9. From 145.00 take 76-84. 10. From 14.674 take 5.91. 11. From 761.8109 take 18.9118. 12. From 171.195 take 125.9176. 13. From 480 take 245.0075. 14. 3.024 x 2.23 15. 25.238 x 12.17. 16. •007853 x .035 = .000274855. 17. •007 x .0008. 18. 25•238 x 12-17. 19. 84179 x .0385. 20. 4:18000 : 1812. 21. 186513.239 • 304.81. 22. Divide 7.25406 by :957. Ans. 7.58. 27 ig35, 2'so, and 3 to decimals. 34. Reduce 7 cwt. 3 qrs. 17 lb. 10 oz. 12 drs. to the decimal of a ton. 35. Reduce 8 feet, 6 inches to the decimal of a mile. 20 27 EVOLUTION. EVOLUTION, or the extraction of roots, is to find such a number as being multiplied into itself a certain number of times will produce that number: if we resolve 36 into two equal factors, namely, 6 and 6, each of these equal factors is called a root of 36, because 6 x 6 36, and 6 is the square root of 36. And 27, resolved into three equal factors, 3, 3, and 8, each factor is called a root of 27, because 3 x 3 x 3 27, and 3 is the 3d or cube root of 27—and the same of other numbers. A square number cannot have more places of figures than double the places of the root, and but one less. A cube cannot have more figures than triple the number of the root, nor but two less. SQUARE ROOT. RULE.—1. Separate the given number into periods of two figures, beginning with units. 2. Find the root of the period on the left, and place it in the quotient, and its square under said period, which subtract from the number above. 3. Then bring down the next period, (two figures, and place it on the right of the remainder, as in Division, and this forms a new dividend. 4. Now double this figure, or root, in the quotient, and place it on the left of the new dividend for a divisor. 5. Then consider how often the divisor is contained in the dividend, omitting the right-hand figure, and place the result on the right of the root in the quotient, and then place this figure on the right of the number produced by doubling for a divisor, and multiply as in Division till the root of all the periods is extracted. FOR DECIMALS.—When decimals occur in the given number, they must be pointed both ways from the decimal point, and the root must consist of as many figures, of whole numbers and decimals respectively, as there are periods of integers or decimals in the given number. When a decimal alone is given, annex one cipher, if necessary, so that the number of decimal places shall be equal; and the number of decimal places in the root will be equal to the number of periods in the given decimal. FOR VULGAR FRACTIONS.—1. Reduce mixed numbers to improper fractions, and compound fractions to simple ones, and then reduce the fraction to its lowest terms. 2. Extract the square root of the numerator and denominator separately, if they have exact roots; but if they have not, reduce the fraction to a decimal, and then extract the root, as above, &c. PROOF.-Square the root, and add in the remainder. CUBE ROOT. RULE.—1. Separate the given number into periods of three figures each, placing a point over units, then over every third figure towards the left in whole numbers, and over every third figure towards the right in decimals. 2. Find the greatest cube in the first period on the left hand; then placing its root on the right of the number for the first figure of the root, subtract its cube from the period, and to the remainder bring down the next period for a dividend. 3. Square the root already found, giving it its true local value; multiply this square by 3, and place the product on the left of the dividend for a divisor; find how many times it is contained in the dividend, and place the result in the root. 4. Multiply the root already found, regarding its local value by this last figure added to it, then multiply this product by 3, and place the result on the left of the dividend under the di. visor; under this result write also the square of the last figure placed in the root. 14 DUODECIMALS, OR CROSS-MULTIPLICATION. 5. Finally, add these results to the divisor; multiply the sum by the last figure placed in the root, and subtract the product from the dividend. To the right of the remainder bring down the next period for a new dividend; find a new divisor, and proceed with the operation as above. 6. When there is a remainder, periods of ciphers may be annexed. NOTE.-If the right-hand period of decimals is deficient, this deficiency must be supplied by ciphers. The root must contain as many places of decimals as there are periods of decimals in the given number. The cube root of a vulgar fraction is found by extracting the root of its numerator and denominator, or reducing the fraction to a decimal. A mixed number should be reduced to an improper fraction. PROOF.—Multiply the root into itself twice; add the remainder. QUESTIONS. 1. What is the square root of 54590-25 ? Ans. 2345. 2. What is the square root of 3271.4007 ? Ans. 57:19+ 3. What is the square root of 96385163? Ans. 9817+ 4. What is the square root of 10342656 ? Ans. 3216. 5. What is the square root of 964.5192360241 ? Answer. 31.05671. 6. What is the square root of .0000316969 ? Ans. •00563. 7. What is the square root of 18, 148, 61, 5216 8. What is the cube root of 259694072 ? Ans. 638. 9. What is the cube root of 34328125 ? Ans. 325. 10. What is the cube root of 37862135? Ans. •723+ 11. What is the cube root of 34965783? Ans. 327. 12. What is the cube root of 125 ? 13. What is the cube root of 5 or •018115942 ? 14. What is the cube root of ? 15. What is the cube root of ¥ 435 125 ? 16. What is the cube root of 1896 ? Ans.. 272 Note. This part of Arithmetic is more fully explained in the “ Columbian Calculator." DUODECIMALS, OR CROSS-MULTIPLICATION. This rule is highly valued by artificers and workmen, particularly carpenters and joiners, in measuring and estimating the value of their work. The dimension being taken in feet, inches, and twelfths. A foot is divided into 12 parts, called inches, |