Ex. 1.,, 1, 70. 2. 5, 31, 47's, 7377. 3. 3, 4, 52. 900 4. 4, 49, 53188, 53,997. 5. 3.1111, 42.7666, 0.1357. 6. 67.3454545, 8.6216216, 0.2424242, 0.8888888. 7. 69.1946. (8.) '81.01492. (10.) 71.063218639. 11. 3.7 and 4.62. (12.) 3.5 and 3.52. 14. 9.194 and 19.98379015. 15. 98.16 and 12.916. 16. 133.256 and 25.664246913580. DUODECIMALS. Duodecimals are the parts resulting from the continued separation of the integer, and of the parts produced, into 12 equal parts: thus, the separation of a foot into 12 inches, and of each inch into 12 parts, is a duodecimal separation. The first parts produced from the integer are called primes, and the next seconds, &c. The most common, if not the only, application* of duodecimals, is in the measurement of surfaces and solids, when the dimensions are taken in feet and inches. DUODECIMAL MULTIPLICATION. Rule 1. Multiply by the number of the units or feet, and take parts out of unity or 1 foot for the lower denominations. Rule 2. Multiply the whole of the multiplicand, 1 st, by the number of the units or feet, as in common multiplication; 2 ndly, by the number of the 12 ths or inches, and consider each product as one step lower in value; or the product of inches consider as 12 ths of an inch, and the product of feet as inches; 3 rdly, by the number of 12 ths of inches, if any, and consider each product as 2 steps lower in value; and so on, if there are any lower denominations employed in the multiplier. Rule 3. called Cross Multiplication. Multiply each term in the multiplicand separately by each term in the multiplier. And observe, as above, that when the multiplier is units, or a number of feet, the product will be of the same value as its multiplicand; when the multiplier is primes, or a number of inches, each product will be one step lower in value; when seconds, or twelfths of inches, 2 steps, &c., and that each product in a less denomination than feet, is, if it will admit of it, to be reduced into a higher denomination. * In consequence of this application, in speaking of the multiplication resulting from these dimensions, instead of the proper mode of expressing the multiplier by times and parts, it is usual to call them feet and inches; as for taking 3 times and 5-12 ths, to direct the multiplication by 3 feet and 5 inches. EXAMPLE. To multiply 17 feet 10 in. by 9 and 7-12 ths, or 9 feet 7 inches.* In the first method, we multiply 17 feet 10 in. by 9, and take parts for 7-12 ths, saying 6 are the half, and 1 is the sixth of 6. In the second, we multiply, 1 st by 9, and next by 7, taking 1-12 th of each of the latter products by reckoning each product as 1-12 th of its usual value: thus, 7 times 10 are 70, 70-12 ths of an inch are 5 inches and 10-12 ths, carry 5, 7 times 17 are 119 and 5 are 124; 124 inches are 10 feet 4 in. In the third, we multiply, 1 st, 17 feet by 9, of which the product is 153 feet; next 10 inches by 9, of which the product is 7 feet 6 in.; then 17 by 7, reckoning the product, 119, as inches, or 9 feet 11 in.; lastly 10 by 7, reckoning the product as 70-12 ths of an inch, or 5 in. 10-12 ths. N. B. In working the above Exercises it must be recollected, that to say we multiply by 4 feet 6 inches, means only to take 4 times and 6-12 ths, or 4 times; for speaking strictly, we cannot say we multiply by 4 feet, that is, to say we take 4 feet of times. APPLICATION OF DUODECIMALS. In the measurement of surfaces the contents are usually giver in square feet and 12 ths, which are called inches, although as 144 square inches make 1 square foot, each 12 th is properly 12 square inches. Measurements are also made in yards of 9 square feet; in Carpenters' squares of 100 sq. ft. and in Bricklayers' square rods of 272 sq. ft. Land measure is taken in acres, Roods, &c. Solids are measured in cubic inches, cubic feet, cubic yards, and tons of 40 cubic feet. The principles upon which measures in square feet and 12 ths of rectangular surfaces are obtained, are these; taking the length as so many square feet and 12 ths as there are feet and inches in the length, we have the superficial contents for 1 foot in breadth ; and then multiplying these by the number of feet and inches in the given breadth, we obtain the whole superficial contents. The solidity of rectangular solids is obtained from considering the number of square feet, &c. obtained from the length and breadth, as so many cubic feet, &c., and multiplying them by the number of feet and inches in the depth. Hence the common rule is: Multiply the length by the breadth for the surface and that product by the depth for the solidity. EXERCISES. Ex. 1. What are the superficial contents of the floor of a room, 18 ft. 7 in. long, and 11 ft. 10 in. wide? Answer Sq. ft. 219 10 10. Ex. 2. What are the superficial contents in square yards of the floor of a room 23 ft. 4 in. long, and 13 ft. 6 in. wide? Answer 315 sq. ft. 0 in. or 35 yds. 0 ft. 0 in. Ex. 3. A wall is in length 95 ft. 10 in., and in height 11 ft. 6 in., how many square rods of 272 sq. ft. are there, and what is the amount at £ 24 10 per rod of the standard thickness of 1 bricks, the thickness of the wall being 2 bricks? Answer Surf. 4 rods 14 ft. Cont. 5 rods 109 ft. Ex. 4. How many cubic yards are there in a solid of these dimensions, viz. length 18 ft. 4 in., breadth 7 ft. 10 in., depth 9 ft. 6 in.? Answer 50 yds. 14 ft. 3 in. Ex. 5. The dimensions of a case are, length 8 ft. 6 in., breadth 5 feet. 7 in., depth 5 ft. 4 in., what are the cubic contents, and how much is the amount of the freight at 35 s. per ton of 40 cubic feet? Answer Contents 253 cubic ft. 1 in. Amount £ 11 15 INVOLUTION. Involution is the formation of a series of numbers, by the multiplication of the given number and of each successive product by that number. The first product is called the 2 nd power or square; the second product the 3rd power or cube; the third product the 4 th power or biquadrate; and the succeeding products are called the 5 th, 6 th, &c. powers. Thus with the number 4. 41 or the 1 st power is 42 or the 2 nd power or square 43 or the 3 rd power or cube 4 or the 4 th power or biquadrate 4 16 64 256 = 1024 The first power, or the given number, is called the root, and the number of the power is called the index of that power. The corresponding number to any power may be found from the multiplication of the numbers corresponding to those powers, which by addition of their indices make up the index of the given power-thus, the 5 th power of 4 may be formed from the multiplication of the numbers corresponding to the 2 nd and 3 rd powers of 4, as 16 X 64 = 1024. Fractional and decimal numbers are involved in the same manner, and their powers possess the same properties as whole numbers. Thus 'So also 0.4 as a root or 1 st power, has Table of the first 6 powers of single figures. 4 th 5 th 6 th 1 16 81 256 625 1296 2401 1 32 243 1024 3125 7776 16807 32768 59049 1 64 729 4096 15625 46656 117649 262144 531441 4096 6561 |