called inches, or primes, marked thus, ('). Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds, ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds, (""); each third irto twelve equal parts, called fourths, (); and so on to any extent. In this way of dividing a foot, it is obvious, that 1" fourth is 1 fifth is of 1 of 1 of 12, = 20736 of a foot. of th of th of th of 2,248832 of a foot, &c. = Duodecimals are added and subtracted in the same manner as compound numbers, 12 of a less denomination making 1 of a greater, as in the following Note. The marks,,", "", "", &c., which distinguish the different parts, are called the indices of the parts or denominations. MULTIPLICATION OF DUODECIMALS. Duodecimals are chiefly used in measuring surfaces and solids. 1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide? is to take of 16=48, that is, 48'; and the 1' which we reserved makes 49', 4 feet 1'; we therefore set down the 1', and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answer, 20 feet, 8', and 9". The only difficulty that can arise in the multiplication of duodecimals is, in finding of what denomination is the product of any two denominations. This may be ascertained as above, and in all cases it will be found to hold true, that the product of any two denominations will always be of the denomination denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of 7' x 3' is "; consequently, the product is 21"; and thus primes multiplied by primes will produce seconds; primes multiplied by seconds produce thirds; fourths multiplied by fifths produce ninths, &c. It is generally most convenient, in practice, to multiply the multiplicand first by the feet of the multiplier, then by the inches, &c., thus :- 2. How many solid feet in a block 15 ft. 8' long, 1 ft. 5' wide, and 1 ft. 4' thick? From these examples we derive the following RULE :Write down the denominations as compound numbers, and in multiplying remember, that the product of any two denominations will always be of that denomination denoted by the sum of their indices. EXAMPLES FOR PRACTICE. 3. How many square feet in a stock of 15 boards, 12 ft. & in length, and 13' wide? Ans. 205 ft. 10'. 4. What is the product of 371 ft. 2' 6" multiplied by 181 ft. 1 9/"? Ans. 67242 ft. 10' 1" 4" 6. Note. Painting, plastering, paving, and some other kinds of work, are done by the square yard. If the contents in square feet be divided by 9, the quotient, it is evident, will be square yards. 5. A man painted the walls of a room 8 ft. 2' in height, and 72 ft. 4 in compass; (that is, the measure of all its sides;) how many square yards did he paint? Ans. 65 yds. 5 ft. 8' 8". 6. There is a room plastered, the compass of which is 47 ft. 3', and the height 7 ft. 6'; what are the contents ? Ans. 39 yds. 3 ft. 4′ 6′′. 7. How many cord feet of wood in a load 8 feet long, 4 feet wide, and 3 feet 6 inches high? Note. It will be recollected, that 16 solid feet make a cord foot. Ans. 7 cord feet. 8. In a pile of wood 176 ft. in length, 3 ft. 9' wide, and 4 ft. 3' high, how many cords? Ans. 21 cords, and 7 cord feet over. 9. How many feet of cord wood in a load 7 feet long, 3 feet wide, and 3 feet 4 inches high? and what will it come to at $40 per cord foot? Ans. 48 cord feet, and it will come to $175. 10. How much wood in a load 10 ft. in length, 3 ft. 9' in width, and 4 ft. 8' in height? and what will it cost at $1'92 per cord? Ans. 1 cord and 21 cord feet, and it will come to $2'624. ¶ 104. Remark. By some surveyors of wood, dimensions are taken in feet and decimals of a foot. For this purpose, make a rule or scale 4 feet long, and divide it into feet, and each foot into ten equal parts. On one end of the rule, for 1 foot, let each of these parts be divided into 10 other equal parts. The former division will be 10ths, and the latter 100ths of a foot. Such a rule will be found very convenient for surveyors of wood and of lumber, for painters, joiners, &c.; for the dimensions taken by it being in feet and decimals of a foot, the casts will be no other than so many operations in decimal fractions. 11. How many square feet in a hearth stone, which, by a rule, as above described, measures 4'5 feet in length, and 2'6 feet in width? and what will be its cost, at 75 cents per square foot? Ans. 117 feet; and it will cost $8'775. 12. How many cords in a load of wood 7'5 feet in length, 3'6 feet in width, and 4'8 feet in height? Ans. 1 cord,1ft. 13. How many cord feet in a load of wood 10 feet long, 3'4 feet wide, and 3'5 feet high? Ans. 76. QUESTIONS. 1. What are duodecimals? 2. From what is the word derived? 3. Into how many parts is a foot usually divided, and what are the parts called? 4. What are the other denominations? 5. What is understood by the indices of the denominations? 6. In what are duodecimals chiefly used? 7. How are the contents of a surface bounded by straight lines found? 8. How are the contents of a solid found? 9. How is it known of what denomination is the product of any two denominations? 10. How may a scale or rule be formed for taking dimensions in feet and decimal parts of a foot ? INVOLUTION. 105. Involution, or the raising of powers, is the multiplying any given number into itself continually a certain number of times. The products thus produced are called the powers of the given number. The number itself is called the first power, or root. If the first power be multiplied by itself, the product is called the second power or square; if the square be multiplied by the first power, the product is called the third power, or cube, &c.; thus, 5 is the root, or 1st power, of 5. =5a =58. 5×5 25 is the 2d power, or square, of 5, 5X5X5 125 is the 3d power, or cube, of 5, 5×5×5×5—625 is the 4th power, or biquadrate, of 5, =5*, S The number denoting the power is called the index, or exponent; thus, 5* denotes that 5 is raised or involved to the 4th power. 1. What is the square, or 2d power, of 7? 2. What is the square of 30? 3. What is the square of 4000 ? 4. What is the cube, or 3d power, of 4 ? 5. What is the cube of aoo? 6. What is the 4th power of 60 ? 7. What is the square of 1? of 4? 8. What is the cube of 1? of 4? 9. What is the square of ? 10. What is the cube of. ? 11. What is the square of? 12. What is the square of 1'5? 13. What is the 6th power of 1'2?. 14. Involve 24 to the 4th power. Note. A mixed number, like the Ans. 49. Ans. 900, Ans. 16000000. Ans. 64. Ano. 512000000. Ans. 12960000.. of 3? of 2? above, may be reduced to an improper fraction before involving: thus, 24 = 2; or it may be reduced to a decimal; thus, 242′25. 15. What is the square of 47? Ans. 55612518. 동 = Ans. 16212311. 16. What is the value of 7, that is, the 4th power of 7? The powers of the nine digits, from the first power to the fifth, may be seen in the following Squares for 2d Powers1 | 4 9 jor 3d Powers[18 27 64 125 | 216 2401 4096 6561 16 25 36 49 64 81 343 512 729 |