or 36'- 3 feet. When feet are multiplied by seconds, the product is in seconds: thus 6 feet mu'tiplied by 6 seconds, that is, of a foot by 12 of of a foot, the product is of a foot, or 36"-3 inches. 36 8 48 When inches are multiplied by inches, the product is in seconds. Thus, 6 inches multiplied by 8 inches, that is, 12 of a foot by of a foot, the product is 144 of a foot, or 48": 4 inches. When inches are multiplied by seconds, the product is in thirds. Thus, 6 inches multiplied by 8 seconds, that is, of a foot by 1 of 12 of a foot, the product is 142 of a foot, or 48"-4 seconds. When seconds are multiplied by seconds, the product is in fourths. Thus, 6" multiplied by 8", that is, 1 of 1 12 12 of a foot, by of of a foot, the product is 20736 of a foot, or 4 4 thirds. 1 48 1 12 12 This method of showing the denomination of the product resulting from the multiplication of duodecimals by duodecimals may be extended to any number of places whatever; but sufficient has been said, to show that the product is always of that denomination denoted by the sum of the indices of the two factors. Feet multiplied by feet, produce feet. Feet multiplied by primes, produce primes. &c. Primes multiplied by primes, produce seconds. &c. Seconds multiplied by seconds, produce fourths &c. Thirds multiplied by thirds, produce sixths. &c If we would find the square feet in a floor 6f. 4′ 8 in length, and 4f. 6′ 5′′ in breadth, we should proceed as. follows 6 f. 4' 8" 2' 7" 11" 4"" 3 2′ 4′′ 0"". 25 6' 8". 28 f. 11′ 7′′ 11′′ 4""" We began on the right hand, and multiplied the whole multiplicand, first by the seconds in the multiplier, then by the inches, and lastly by the feet, and added the results together, and thus obtained the answer. That the above answer is the true one, will appear very clearly from the following considerations. The 8 seconds, as we have already shown, may be considered in relation to feet as 144, and the 5 seconds as 144, the product of which is 20736 of a foot, or 40' which is equal to 3"" and 4""; writing down the 4' 4 1728 6 8 40 144 48 we reserve the 3"" to be added to the product of 4' by 5". 4 being 12 of a foot, and 5" being of a foot, their product is 20 of a foot, or 20′′, to which adding the 3", that were reserved, we had 23"", equal to 1" and 11"; we wrote down the 11", and reserved the 1" to be added to the product of 6 feet by 5". 6 feet being of a foot, and 5" being of a foot, their product is of a foot, or 30", to which we added the 1" reserved, and thus had 31", equal to 2' and 7", both of which we wrote down. Having completed the multiplication by the seconds, we next multiplied by the inches: 8" being 144 of a foot, and 6' being 12 of a foot, their product is 1728 of a foot, or 48" 4"; we therefore put down a cipher in the place of thirds, and reserved the 4" to be added to the product of inches by inches. 4 inches being 12 of a foot, and 6 inches of a foot, their product is 1444 of a foot, or 24′′, to which we added the 4" reserved, making 28′′—2′ and 4"; writing down the 4", we reserved the 2' to be added to the product of feet by inches. 6 feet being of a foot, and 6 inches of a foot, their product is of a foot, 331920 or 36', to which we added the 2' reserved, making 38 3 feet and 2 inches, both of which we wrote down Lastly, we multiplied by the feet in the multiplier. S". or 144 of a foot being multiplied by 4 feet, or 4 of a foot, their product is 324 of a foot, or 32" 2' and 8"; setting down the 8", we reserved the 2' to be added to the product of inches by feet. 4', or of a foot being multiplied 8 144 by 4 feet, or of a foot, their product is 11⁄2 of a foot, or 16', to which we added the 2' reserved, making 18'=1 foot and 6 inches; writing down the 6', we reserved the 1 foot to be added to the product of feet by feet. 6 feet being multiplied by 4 feet, their product is 24 feet, to which we added the 1 foot reserved, making 25 feet. By adding these three partial products together, we obtained the answer to the question. Therefore, to multiply one number consisting of feet, inches, seconds, &c. by another of the same kind, we give the following rule. RULE. Place the several terms of the multiplier under the corresponding ones of the multiplicand. Beginning on the right hand, multiply the several terms of the mul tiplicand by the several terms of the multiplier successively, placing the right hand term of each of the partial products under its multiplier; then add the partial products together, observing to carry one for every twelve, both in multiplying and adding. The sum of the partial products will be the answer. Questions in duodecimals are very commonly performed by commencing the multiplication with the highest denomination of the multiplier, and placing the partial products as in the first of the two following operations. The result is the same, whichever method is adopted. The second operation, however, is according to the rule we have given, and is more conformable to the multiplication of numbers accompanied by decimals. When there are not feet in both the factors, there may not be any feet in the product; but, after what has been said, there will be no difficulty in determining the places of the product. 1. Multiply 14f. 9' by 4f. 6'. 2. What are the contents of a marble slab, whose length is 5 f. 7', and breadth 1 f. 10'? 3. How many square feet are there in the floor of a hall 48f. 6' long, and 24f. 3' wide? 4. Multiply 4f. 7 8" by 9f. 6'. 5. How many square feet are there in a house lot 43f. 3' in length, and 25 f. 6' in breadth > 6. What is the product of 10f. 4 5" by 7f. 8' 6"? 7. Calculate the square feet in an alley 44f. 2′ 9′′ long, and 2f. 10' 3" 2" 4" wide. 8. How many square feet are there in a garden 39 f. 10' 7" long, and 18f. 8' 4" wide ? 9. What is the product of 24f. 10 8" 7" 5" by 9 f. 4'6"? 10. Compute the solid feet in a wall 53 f. 6' long, 12f. 3' high, and 2f. thick. 11. The length of a room is 20 feet, its breadth 14 feet 6', and its height 10f. 4'. How many yards of painting are there in its walls, deducting a fire place of 4f. by 4f. 4 and two windows, each 6f. by 3f. 2'? 12. How many solid feet in a pile of wood 22f. 6' long, 12f. 8' wide, and 5f. 8' high? 13. How many yards of plastering in the top and four walls of a hall 58f. 8' long, 21f. 4' wide, and 13f. 9 high; deducting for 2 doors each 7f. 6' high and 4f. wide; for 7 windows each 6 f. 2' high, and 3f. 10' wide; for 2 fire places, each 3f. 6' high, and 4f. wide, and for a mop board 9 inches wide around the hall ? 14. How many yards of papering in a room 17f. 8 long, 16f. 9' wide, and 12f. 6' high; deducting for 2 doors each 6f. 6' high, and 4f. wide; for a fire place 4f. 6' high and 3 f. 10' wide; for 3 windows each 5f. 6' high and 3f. 8′ wide, and for a mop board 8 inches wide around the room? 15. How many yards of carpeting, yard wide, will be required for a room 21 f. 6' long, and 18 ft. wide? 16 What will the plastering of a ceiling come to, at 10 cents a square yard, supposing the length 21 feet 8 inches, and the breadth 14 feet 10 inches? XXVIII. INVOLUTION. INVOLUTION is the multiplication of a number by itself. The number, which is thus multiplied by itself is called the root. The product, which we obtain by multiplying a number by itself, is called a power of that number. The number involved is itself the first power, and is the root of all the other powers. A number, multiplied once by itself, is said to be involved or raised to the second degree, or second power; multiplied again, to the third degree, or third power; and so on. For example, 3×3 is raised to the second power of 3, which is 9; 3X3 X 3 is raised to the third power of 3, which is 27; &c. We distinguish the powers from one another by the number of times, that the root is used as factor in the multiplication of itself. Thus, 3x3 produces the second power of 3, because 3 is used twice as factor; 3x3x3 produces the third power of 3, because 3 is used three times as factor; 3X3X3X3 produces the fourth power of 3, because 3 is used four times as factor; and so on. 9 81 27 A fraction is involved in the same manner by multiplying it continually into itself; thus, the second power of 3, is 2×21; the third power is the fourth power is X-236; and so on. So also in decirnals the second power of .2, is .2 X.2=.04; the third power is .04 .2 .008; the fourth power is .008 X.2 =.0016; and so on. To involve a mixed number, reduce it first to an improper fraction, or the vulgar fraction to a decimal, and then involve it. Thus, 1 when reduced to an improper fraction, is, the second power of which 24; the third power is 27=33; &c. If, instead of reducing 11 to an improper fraction, we reduce the vulgar fraction to a decimal, we have 1.5, the second power of which is 2.25; the third power, 3.375; &c. |