As a mere introduction to the use which the intelligent stuhereafter make of this principle, we here insert a few dent may n an improper fraction, and m 24. 17 0 16 n nQ m 337. Let Q be any quantity, m = d. Then (77) n = m+d: consequently mQ + dq m do Q+ ; that is, Q is increased in a ratio m 1 338. When n— m = 1, the quantity Q is increased by th m Q+. Thus 24 X=24+3=27. m Also, 24 yds. at $1,121 per yd. = $24+ $3 = $27. Here we have added to $24, the cost of 24 yds. at $1, the cost of 24 yds. at $1, which is $3. Proof. 112 cts.. 2688 12 $27,00 Required the cost of 294 yds. at $1,331 per yd. Here the supplement is 33 cts. $1. We therefore add to 291, considered as $294, its third part, thus: 291 + 93 $39, Änswer. Here, as it is of no consequence which number we take for multiplicand, we add to $1,663 its fourth part, which is 413, and have 1,663 +413 $2081 or $208,331, for the answer. 1,66 × 1000 Proof. 8 = 1666,663_208,331. See art. 273. 8 339. This rule is used in measuring cord-wood, artificer's work, &c. The foot is the principal measure, and is divided as follows: 1 foot, ft. is equal to 12 inches, or primes, marked thus,' (1) 1 inch..... 1 second. 1 third,...... 12 parts, or seconds.. .12 thirds.. 12 fourths The inch, or prime, is 12 of 1=141 of a foot. 12 of a foot, &c. These parts, from their 1728 G2 of a foot. The second is (1), or The third is (1), or 11⁄2 of 114 — regular ratio of 12, are called duodecimals, which name is derived from the Latin duodecim, twelve. Multiplication of Duodecimals. 340. From the nature of fractions, we shall easily perceive what parts will be produced in multiplying together any of the above denominations. For example, if we multiply by feet, as these are integers (whole numbers,) having a unit for denominator, it is evident that the denominator of the part multiplied will not be altered; thus, 3 feet multiplied by 2 feet, will give 6 feet; 3 in. multiplied by 2 ft., that is, X, or 6 in. Again, 3 sec. multiplied by 2 ft., or 1X1, or 6 seconds, etc. 3 129 If we multiply by inches, as the denominator of these is 12, the denominator of the part multiplied will become the same as that of the next lower denomination; thus, 3 ft. multiplied by 2 in. will give 6 in.; that is, X6, or 6 inches; 3 in. multiplied by 2 in.; that is, 12X12 144, or 6 seconds; 3 sec. multiplied by 2 in., or 14412=1725, or 6 thirds, &c. From what has been said, the products of any other parts will easily be estimated by one who has learned fractions. 341. As the second is the square of the prime, the third, the cube, &c., if we consider the feet as having 0 for their index, the inches 1, the seconds 2, the thirds, 3, &c., if we multiply these together, the parts produced will have for their index the sum of the indices of the parts multiplied; thus, the index of the prime is 1; that of the second is 2, and the sum 3, of these indices, is the index of the parts produced when we multiply inches by seconds. It is the same for any other parts, seeing that their product contains the factor, as often as its index contains a unit. For example, to multiply 7 ft. 3 in. 4 sec. by 2 ft. 4 in., we write the numbers thus: and first multiply all the parts of the multiplicand by the 2 ft. of the multiplier. Beginning with the lowest denomination, which is here seconds, we say 2° × 4′′ = 8′′, which we place under seconds; then 2° 3' 6', which we write under inches lastly, 2° × 7° — 14°, which we write under feet. Then, to multiply by 4', we say 4' x 4" 16"" 1"+4""; we therefore write 4"" one place farther to the right, and retain the 1", to add to the next product. Then 4' X 3'12", and, adding the 1" retained, we have 13" 1'+1"; we therefore write 1" under 8", and retain the 1' to add to the next product. Lastly, 4' X 7°28', and 1' retained, is 29'2° 5'; we therefore write 5' under the primes or inches, and the 2 ft. under feet. Having added the two products, we have 16 feet, 11 primes, 9 seconds, 4 thirds, for the total product. 342. We have seen (110) that the content of any superficies is estimated in squares, and is the product of the length and breadth, each taken in the same measure. Artificers' work, such as glazing, painting, plastering, paving, roofing, &c., is usually paid by the square yard or square foot; and, as the dimensions are taken in feet and inches, being generally measured with a carpenter's rule, such work is conveniently calculated by duodecimals. Parts less than seconds are commonly omitted in practice. Suppose the roof of a green-house to be 100 ft. 9 in. long and 21 ft. 6 in. wide, what is the amount of glazing, and what will it cost at 9 cts. per square foot? = 12 50 4 6" 2166° 1' 6" 18 The scholar must take particular notice, that in the product, the prime 1' is of a square foot, or of 144 square inches, and is consequently 12 square inches; that the 6", being a number of hundred and forty-fourths, is 6 square inches, and hence, that 1' 6" √2+8=4=; or, recollecting 144 1; that 11 d. of a shilling, he will see that I' 6" or 11⁄2 twelfths of a s. f. of a s. f. Therefore, 2166° 1' 6" 2166. s. f.; which, as dollars and cents are decimals, we may express thus: 2166,125 s. f. Then, multiplying by 9, we have $194,95125 or $194,95 for the required cost. Examples. = 1. Suppose a room to be 47 ft. 2 in. long, and 13 ft. 6 in. wide; what will the ceiling of it cost, at 25 cts. per square yard? Ans. $17,683 2. How many yards of carpeting, that is yard-wide, will carpet a room that is 36 ft. 8 in. long, and 25 ft. 6 in. wide, and what will it cost, at $1,124 cts per yard? Ans. 1038 yds., and the cost $116,871. 3. How many square yards of paving are there in a courtyard, which is 77 ft. 3 in. long, and 66 ft. 8 in. wide; how many bricks will pave it, allowing each brick, when laid, to cover a |