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. 3' in length, and 13' wide? Ans. 205 ft. 10'. 4. What is the product of 371 ft. 2' 6" multiplied by 181 ft. 19/? 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. 4 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 $2'621 ¶ 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 75 feet in length, 3'6 feet in width, and 4'8 feet in height? Ans. 1 cord, 1 ft. 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. 5X5 25 is the 2d power, or square, of 5, 5X5X5=125 is the 3d power, or cube, of 5, =52 =5o. 5X5X5X5 625 is the 4th power, or biquadrate, of 5, 54. 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 800 ? 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. Ans. 49. Ans. 900. Ans. 16000000. Ans. 64. Ans. 512000000. of 2 ? of 3? Note. A mixed number, like the above, may be reduced to an improper fraction before involving: thus, 24 = 1; or it may be reduced to a decimal; thus, 24 = 2'25. 15. What is the square of 47? Ans. 612518. 16. What is the value of 74, that is, the 4th power of 7? Ans. 2401. The powers of the nine digits, from the first power to the fifth, may be seen in the following jor 3d Powers 1 | 8| 27 64 125 216 343| 512 729 Biquadrates or 4th Powers 1 15 81 256|€25|1996|2401| 4096|6561 Sursolids |or 5th Powers|1 |32|243|1024|3125 |7776|16807 |32768 (59049 EVOLUTION. T 106. Evolution, or the extracting of roots, is the method of finding the root of any power or number. The root, as we have seen, is that number, which, by a continual multiplication into itself, produces the given power. The square root is a number which, being squared, will produce the given number; and the cube, or third root, is a number which, being cubed or involved to the 3d power, will produce the given number: thus, the square root of 144 is 12, because 122 = 144; and the cuhe root of 343 is 7, because 73, that is, 7 X 7 X 7, = 343; and so of other numbers. Although there is no number which will not produce a perfect power by involution, yet there are many numbers of which precise roots can never be obtained. But, by the help of decimals, we can approximate, or approach, towards the root to any assigned degree of exactness. Numbers, whose precise roots, cannot be obtained, are called surd numbers, and those, whose roots can be exactly obtained, are called rational numbers. The square root is indicated by this character placed before the number; the other roots by the same character, with the index of the root placed over it. Thus, the square root of 16 is expressed 16; and the cube root of 27 is expressed 27; and the 5th root of 7776,5/7776. When the power is expressed by several numbers, with the sign + or - between them, a line, or vinculum, is drawn from the top of the sign over all the parts of it; thus, the square root of 21 - 5 is / 21 5, &c. EXTRACTION OF THE SQUARE ROOT. 107. To extract the square root of any number is to find a number, which, being multiplied into itself, shall pro-" duce the given number. 1. Supposing a man has 625 yards of carpeting, a yard wide, what is the length of one side of a square room, the floor of which the carpeting will cover? that is, what is one side of a square, which contains 625 square yards? We have seen, (T 35,) that the contents of a square surface is found by multiplying the length of one side into itself, that is, by raising it to the second power; and hence, having the contents (625) given, we must extract its square root to find one side of the room. This we must do by a sort of trial: and, 1st. We will endeavour to ascertain how many figures there will be in the root. This we can easily do, by pointing off the number, from units, into periods of two figures each; for the square of any root always contains just twice as many, or one figure less than twice as many figures, as are in the root; of which truth the pupil may easily satisfy himself by trial. Pointing off the number, we find, that the root will consist of two figures, a ten and a unit. 2d. We will now seek for the first figure, that is, for the tens of the root, and it is plain, that we must extract it from the left hand period 6, (hundreds.) The greatest square in 6 (hundreds) we find, by trial, to be 4, (bundreds,) the root of which is 2, (tens, 20;) therefore, we set 2 (tens) in the root. The root, it will be recollected, is one side of a square. Let us, then, form a square, (A, Fig. I.) each side of which shall be supposed 2 tens, 20 yards, expressed by the root now obtained. The contents of this square are 20 × 20 400 yards, now disposed of, and which, consequently, are to be deducted from the whole number of yards, (625,) leaving 225 yards. This deduction is most readily performed by subtracting the square number 4, (hundreds,) or the square of 2, (the figure in the root already found,) from the period 6, (hundreds,) and bringing down the next period by the side of the remainder, making 225, as before. |