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 $8775. 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 11ft. 13. How many cord feet in a load of wood 10 feet long, 3'4 feet wide, and 3'5 feet high? Ans. 776. 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. rumber 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 of 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. 5×5 25 is the 2d power, or square, of 5, =52. 5X5X5=125 is the 3d power, or cube, of 5, 5X5X5X5=625 is the 4th power, or biquadrate, of 5, =54. S =53. The number denoting the power is called the index, or exponent; thus, 54 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. Ans. 12960000. of 3? of 2 ? Note. A mixed number, like the above, may be reduced to an improper fraction before involving: thus, 2 = 1; or it may be reduced to a decimal; thus, 22'25. 15. What is the square of 43? Ans. 62510. Ans. 1821231 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 TABLE. Roots for 1st Powers|1| 2 3 4 5 9 6 71 8 25 36 49 64 81 Cubes |or 3d Powers|1| 8|27| 64| 125| 216 343 512 729 Biquadrates or 4th Powers|1 |16|81| 256 625 1296 2401 4096 6561 Sursolids or 5th Powers 1 |32 |243 |1024 |3125 |7776 |16807 |32768 |59049 EVOLUTION. ¶ 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 rool, 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 122144; and the cube root of 343 is 7, because 73, that is, 7 × 7 × 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 7776,7776. 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 produce 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, ( 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 d 20 OPERATION. 625(2 4 225 FIG. I. A હૈ| ૪૩ 400 20 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, (hundreds,) 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 X 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. 3d. The square A. is now to be enlarged by the addition of the 225 remaining yards; and, in order that the figure may retain its square form, it is evident, the addition must be made on two sides. Now, if the 225 yards be divided by the length of the two sides, (20+20 40,) the quotient will be the breadth of this new addition of 225 yards to the sides c d and b c of the square A. But our root already found, = 2 tens, is the length of one side of the figure A; we therefore take double this root, = 4 tens, for a divisor. 5 yds. OPERATION-CONTINUED. 625(25 4 45)225 3 FIG. II. 20 yds. 20 5 100 5 yds. D d A C 25 5 yds. The divisor, 4, (tens,) is in reality 40, and we are to seek how many times 40 is contained in 225, or, which is the same thing, we may seek how many times 4 (tens) is contained in 22, (tens,) rejecting the right hand figure of the dividend, because we have rejected the cipher in the divisor. We find our quotient, that is, the breadth of the addition, to be 5 yards; but, if we look at Fig. II., we shall perceive that this addition of 5 yards to the two sides does not complete the square; for there is still wanting, in the corner D, a small -square, each side of which is equal to this last quotient, 5; we must, therefore, add this quotient, 5, to the divisor, 40, that is, place it at the right hand of the 4, (tens,) making it 45; and then the whole divisor, 45, multiplied by the quotient, 5, will give the contents of the whole addition around the sides of the figure A, which, in this case, being 225 yards, the same as our dividend, we have no remainder, and the work is done. Consequently, Fig. II. represents the floor of a square room, 25 S* 20 yds. α 20 yds. |