Contracted thus: 3 1',12,8×3×9=91b. Ans. ‚9′ׂ1' ×1×,4′×,8′ 2. If 2 men, in of a year, expend $561, how much money will defray the expences of 6 persons for 5 months, at the same rate? Ans. $300. 3. If a man earn $41 in 82 days, how much would 20 men earn in 100 days, at that rate? Ans. 964D. 70c. 515m. 4. If$100 dollars principal gain $5 interest in 12 months, how much will 840 gain in 2 months? 5. If 23 yards of cloth, is the value of 3 yards, quality? Ans. 43c. 31m. of a yard wide, cost $2%, what of a yard wide, of the same Ans. $2.40 DUODECIMALS,* Are fractions so called because they decrease by twelves; inches being twelfths of a foot, which is the integer; seconds twelfths of an inch; thirds twelfths of a second, and so on; as in the following table. Duodecimals are chiefly useful to ascertain the superfi cial or solid contents of such things as are measured by feet, inches, &c. Addition and Subtraction of Duodecimals are performed as in Compound Addition and Subtraction. This word is derived from the Latin word duodecim, which signifies twelve. MULTIPLICATION OF DUODECIMALS. RULE.* 1. Place the multiplier under the multiplicand, in such a manner that the feet of the multiplier shall stand under the lowest denomination of the multiplicand. 2. Multiply the multiplicand by each term or denomination of the multiplier, separately, as in Compound Multiplication, (always carrying one for every twelve, from each denomination to the next higher,) and place the right hand term of each product under that denomination of the multiplicand by which it is produced. 3. Add together the several partial products, as in Compound Addition, and their sum will be the total product required. Note 1.-If there are no feet in the multiplier, supply their place with a cipher; and if any other denomination between the highest and lowest, either in the multiplier or multiplicand, be wanting, write a cipher in its place. Note 2.-The marks which designate the denominations of duodecimals, are called indices; the index of feet being 0; that of inches 1, that is, one mark; that of the seconds 2 marks, &c. When any two of the denominations are multiplied together, the index of their product is equal to the sum of the indices of the two factors. Thus, if feet be multiplied by feet, the product will be feet; for the sum of the indices is 0+0=0: Ifinches be multiplied by seconds, the product will be thirds; for 1+2=3; &c. In multiplication of duodecimals, the terms of the product are of the same denominations as those terms of the multiplicand which stand over them. EXAMPLES. 1. Multiply 8 feet, 6 inches, 9 seconds, by 6 feet, 5 inches, 7 seconds. *The reason of this rule will be obvious by considering the denominations below the integer as fractional parts of the integer, and then multiplying as in Vulgar Fractions. Thus, feet multiplied by inches give inches; for 2 feet multiplied by 4 inches=2ft.X ft. ft. 8 inches: Inches multiplied by inches give seconds; for 2in. X5in.ft.X ft. 14ft. 12in. 10 seconds; &c. Ft. 8..6. 9 4 11 11..3 .. 9.. 3..6 51..4.. 6 9 Operation. Here, I first place the multiplier so that 6, the feet, stand under 9, the se6.. 5.7 conds (or lowest denomination) of the multiplicand. I then begin with 7 and 9, the lowest denominations of the multiplier and multiplicand, and say 7×9 is 63; which being 5 times 12, and 3 over, I set down 3, and car55..4.. 3.. 8..3 ry 5 to the next denomination. Then, 7x6 is 42, and 5 which I carried makes 47—I set down 11 and carry 3. Then, 7×8 is 56, and 3 which I carried makes 59-I set down 11 and carry 4; and as there is no other term to multiply, I set down the 4 in the next place to the left. In like manner I multiply the multiplicand by the inches, and then by the feet of the multiplier, and set down the right hand term of each product under the term I multiply by. Lastly, I add together the partial products thus found, and the answer is 55ft. 4in. 3" .. 8"' .. 3""". 2. Multiply 14 feet and 2 seconds, by 5 inches, 6 seconds. " Ft. in. 28..2..0 by 24 .. 6 .. 0. PRACTICAL QUESTIONS. 690.. 1..0..0 Note. To find the area, or superficial content, of any parallelogram;* such as a board of equal width from end to end, or the floor of a square room, &c.; multiply the length by the breadth, and the product will be the answer. To find the solid content of any parallelopiped ;* such * See the definitions of these terms in the "Mensuration of Superfices and Solids." as a stick of timber hewn square, of equal bigness from end to end, or a load of wood, &c.; multiply continually together the length, breadth, and height, and the last product will be the answer. 1. What is the superficial content of a board which is 17 feet 7 inches long, and 1 foot 5 inches wide? Ans.* 24sq. ft. 10'.. 11". 2. How many square yards does the floor of a room contain, which is 24 feet 8 inches long, and 17 feet 6 inches wide ? Ans. 47sq. yd. 8ft. 8'. 3. If a stick of hewn timber be 12ft. 10in. long, 1ft. 7in. wide, and 1ft. 9in. thick, how many solid or cubic feet does it contain? Ans.† 35cub. ft. 6'.. 8".. 6""'. 4. How many cubic feet of wood in a load which is 8ft. 6in. long, 2ft. 3in. wide, and 3ft. 9in. high? Ans. 71cub. ft. 8'.. 7"..6" INVOLUTION, Is the raising of powers from any given number, as ä root. The product arising from the multiplication of any given number by itself, is called the second power or square of the number if the second power be multiplied by the said given number, the product is the third power, or cube; if the third power be multiplied by the given number, the product is the fourth power, or biquadrate, &c.; the given number itself being called the first power or root. Thus, 5 is the root, or 1st power of 5. 5X5-25 is the 2d power, or square of 5. In this manner is calculated the following table of the first eight powers of the nine digits. *The inches in this answer are 12ths of a square foot, and the seconds are 144ths of a square foot, or square inches. In this answer, the inches are 12ths, the seconds 144ths, and the thirds 1728ths of a cubic foot. 1123456780 49 343 2401 | 16807 117649 823543 5764801 64 512 4096 32768 262144 209715216777216 9 81 729 656159049 | 531441 4782969/43046721 The index or exponent of any power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So, I is the index or exponent of the first power or root, 2 of the 2d power or square, 3 of the 3d power or cube, &c. Powers, which are to be raised, are usually denoted by writing the index, in small figures, at the upper and right side of the given number: Thus, 52 denotes the 2d power of 5, and 123 denotes the 3d power of 12, &c. If two or more powers be multiplied together, their product is that power whose index is equal to the sum of the indices of the factors, or powers multiplied together: Or,, the multiplication of powers answers to the addition of their indices. Thus, the product of the 2d and 3d powers of any number, is the 5th power of that number; and, if the 2d power be multiplied by itself, the product is the 4th power, &c. PROBLEM. To involve or raise a given number to any proposed power: RULE. Multiply the given number, or first power, continually by itself, till the number of multiplications is 1 less than |