form of a quotient in division; (See Operation 1st.); then, subtracting the square of 30,900 sq. rods,from 1296 sq. rods, leaves 396 sq. rods. The pupil will bear in mind, that the FIG. on the left is in the form of the garden, and contains the same number of square rods, viz. 1296. This figure is divided into parts, called A, B, C, and D. It will be perceived, that the 900 square rods, which we deducted, are found by multiplying the length of A, being 30 rods, by the breadth, being also 30 rods, that is, 302 = 900. To obtain the square rods in B, C, and D, the remaining parts of the figure, we may multiply the length of each by the breadth of each, thus; 30 × 6= 180, 6 x 636, and 30 x 6 180; then 180+36 +180396 square rods; or, add the length of B, that is, 30, to the length of D, which is also 30, making 60; or, which is the same thing, we may double 30, making 60; to this add the length of C, 6 rods, and the sum is 66. Now, to obtain the square rods in the whole length of B, C, and D, we multiply their length, 6 rods, by the breadth of each side, thus, 66 × 6=396 square rods, the same as before. We do the same in the operation; that is, we first double 30 in the quotient, and add the 6 rods to the sum, making 66 for a divisor; next, multiply 66, the divisor, by 6 rods, the width, making 396; then, taking 396 from 396 leaves 0. The pupil will perceive, the only difference between the 1st and 2d operation (which see) is, that in the 2d we neglect writing the ciphers at the right of the numbers, and use only the significant figures. Thus, for 30+6, we write 3 (tens) and 6 (units), which, joined together, make 36; for 900, we write 9 (hundreds). This is obvious from the fact, that the 9 retains its place under the 2 (hundreds). Instead of 60+ 6, we write 66. Omitting the ciphers in this manner cannot reasonably make any difference, and, in fact, it does not, for the result is the same in both. By neglecting the ciphers, we may, perhaps, be at a loss, sometimes, to determine where we must place the square number. In the last example, we knew where the square of the root 3 (tens) 9 (hundreds) should be placed, for the ciphers, at the right, indicate it; but had these ciphers been dropped, we should, doubtless, have hesitated in assigning the 9 its proper place. This difficulty will be obviated by observing what follows. The square of any number never contains but twice as many, or at least but one figure less than twice as many, figures as are in the root. Thus, the square of the root 30 is 900; now, in 900 there are but three figures, and in 30, two figures; that is, the square of 30 contains but one figure more than 30. We will take 99, whose square is 9801, in which there are four figures, and in its root, 99, but two; that is, there are exactly twice as many figures in the square 9801 as are in its root, 99. This will be equally true of any numbers whatever. Hence, to know where to place the several square numbers, we may point off the figures in the given number into periods of two figures each, commencing with the units, and proceeding towards the left. And, since the value of both whole numbers and decimals is determined by their distance from the units" place, consequently, when there are decimals in the given number, we may begin at the units' place, and point off the figures towards the right, in the same manner as we point off whole numbers towards the left. By each of the preceding operations, then, we find that the root of 1296 is 36, or, in other words, the length of each side of the garden is 36 rods. PROOF. This work may now be proved by adding together all the square rods contained in the several parts of the figure, thus : Point off the given number into periods of two figures h, by putting a dot over the units, another over the hundreds, and so on; and, if there are decimals, point them in the same manner, from units towards the right hand. These dots show the number of figures of which the root will consist. Find the greatest square number in the left-hand period, and write its root as a quotient in division; subtract the square number from the left-hand period, and to the remainder bring down the next right-hand period for a dividend. Double the root (quotient figure) already found, and place it at the left of the dividend for a divisor. Write such a figure at the right hand of the divisor, also the same figure in the root, as, when multiplied into the divisor thus increased, the product shall be equal to, or next less than the dividend. This quotient figure will be the second figure in the root. Note. The figure last described, at the right of the divisor, in the second operation, is the 6 rods, the width, which we add to 60, making 66; or, omitting the 0 in 60, and annexing 6, then multiplying 66 by 6, we wrote the 6 in the quotient, at the right of 3, making 36. Multiply the whole increased divisor by the last quotient figure, and write the product under the dividend. Subtract this product from the dividend, and to the remainder bring down the next period, for a new dividend. Double the quotient figures, that is, the root already found, and continue the operation as before, till all the periods are brought down. 18. What is the square root of 470596? A. €86. In this example, we have a remainder, after obtaining one figure in the roos. In such cases, we may continue the operation to decimals, by annexing two ciphers for a new period, and thus continue the operation to any assignable degree of exactness. But since the last figure, in every dividend thus formed, will always be a cipher, and as there is no figure under 10 whose square num ber ends in a cipher, there will, of course, be a remainder; consequently, the pupil need not expect, should he continue the operation to any extent, ever to obtain an exact root. This, however, is by no means necessary; for annexing only one or two periods of ciphers will obtain a root sufficiently exact for almost any purpose. A. 6,3245 + 25. What is the square root of 30? A. 5,4772. 26. What is the square root of ·644? A. &=3. Or, we may reduce the given fraction to its lowest terms before the root is extracted. Thus,√√, Ans., as before. 27. What is the square root of? A. 1§. 28. What is the square root of ? A. . 29. What is the square root of TT? A. TOT. T2 If the fraction be a surd, the easiest method of proceeding will be to reduce it to a decimal first, and extract its root afterwards. 30. What is the square root of Z&? A.,9128 +. 31. What is the square root of A.,9574 +. 32. What is the square root of? A. 83205. 33. What is the square root of 4201? In this example, it will be best to reduce the mixed number to an improper fraction, before extracting its root, after which it may be converted into a mixed number again. A. 20. 34. What is the square root of 912? A. 30. 35. A general has an army of 5625 men; how many must he place in rank and file, to form them into a square? 5625 = 75, Âns. 36. A square pavement contains 24336 square stones of equal size; how many are contained in one of its sides? A. 156. 37. In a circle, whose area, or superficial contents, is 4096 feet, I demand what will be the length of one side of a square containing the same number of feet? A. 64 feet. 38. A gentleman has two valuable building spots, one containing 40 square rods, and the other 60, for which his neighbor offers him a square field, containing 4 times as many square rods as the building spots; how many rods in length must each side of this field measure? 40+60 × 420, Ans. 39. How many trees in each row of a square orchard, containing 14400 trees? A. 120 trees. 40. A certain square garden spot measures 4 rods on each side; what will be the length of one side of a garden containing 4 times as many square rods? A. & rods. 41. If one side of a square piece of land measure 5 rods, what will the side of one measure, which is four times as large? 16 times as large? 36 times a large? A. 10. 20. 30. 42. A man is desirous of forming a tract of land, containing 140 acres, 2 roods and 20 rods, into a square; what will be the length of each side? A. 150 rods. 43. The distance from Providence to Norwich, Conn., is computed to be 45 miles; now, allowing the road to be 4 rods wide, what will be the length of one side of a square lot of land, the square rods of which shall be equal to the square rods contained in said road? A. 240 rods. EXTRACTION OF THE CUBE ROOT, ↑ LXXXVII. Q. Involution, (¶ LXXXIV.,) you doubtless recollect, is the raising of powers; can you tell me what is the 3d power of 3, and what the power is called? A. 27, called a cube. Q. Evolution ( LXXXVII.) was defined to be the extracting the 1st power or roots of higher powers; can you tell me, then, what is the cube root of 27? A. 3. Q. Why? A. Because 3x3 x 3, or, expressed thus, 3327. Q. What, then, is it to extract the cube root of any number? A. It is only to find that number, which, being multiplied into itself three times, will produce the given number. Q. We have seen, (¶ LXXX.,) that, to find the contents of solid bodies, such as wood, for instance, we multiply the length, breadth and depth to gether. These dimensions are called cubic, because, by being thus multiplied, they do in fact contain so many solid feet, inches, &c., as are expressed by their product; but what do you suppose the shape of a solid body is, which is an exact cube? an A. It must have six equal sides, and each side must be exact square. See block A, which accompanies this work. Q. Now, since the length, breadth and thickness of any regular cube are exactly alike, as, for instance, a cubical block, which contains 27 cubic feet, can you inform me what is the length of one side of this block, and what the length may be called? A. Each side is 3 feet, and may be called the cube root of 27. Q. Why? A. Because 33 = 27. Q. What is the length of each side of a cubical block containing 64 cuble inches? A. 4 inches. Q. Why? A. Because 4 X 4 X 4, or 43 64 cubic inches. Q. What is the cube root of 64, then? A. 4. Q. Why? A. Because 43 = 64. Q. What is the length of each side of a cubical block containing 1000 cubic feet? A. 10. Q. Why? A. Because 1031000. 1. In a square box which will contain 1000 marbles, how many will it take to reach across the bottom of the box, in a straight row? A. 10. 2. What is the difference between the cube root of 27 and the cube of 3? 3. What is the difference between 3/8 and 23? A. 6. 4. What is the difference between 3/1 and 13? A. 0. A. 24. 5. What is the difference between the cube root of 27 and the square root of 9? A. 0. 6. What is the difference between 38 and 4? A. 0. Operation by Slate Illustrated. 7. A man, having a cubical block containing 13824 cubic feet, wishes to know the length of each side, without measuring it; what is the length of each side of said block? Should we attempt to illustrate the reason of the rule for extracting the cube root, by exhibiting the picture of the cube and its various parts on paper, it would tend rather to confuse than illustrate the subject. The best method of doing it is, by making several small blocks, which may be supposed to contain a certain proportional number of feet, inches, &c., corresponding with the operation of the rule. They may be made in a few minutes, from a small strip of a pine board, with a common penknife, at the longest, in less time than the teacher can make the pupil comprehend the reason, from merely seeing the picture on paper. In demonstrating the rule in this way, it will be an amusing and instructive exercise, to both teacher and pupil, and may be comprehended by any pupil, however young, who is so fortunate as to have progressed as far as this rule. It will give him distinct ideas respecting the different dimensions of square and cubic measures, and indelibly fix on his mind the reason of the rule, consequently the rule itself. But, for the convenience of teachers, blocks, illustrative of the operation of the foregoing example, will accompany this work. The following are the supposed proportional dimensions of the several blocks used in the demonstration of the above example, which, when put together, ought to make an exact cube, containing 13824 cubic feet: One block, 20 feet long, 20 feet wide, and 20 feet thick; this we will call A Three small blocks, each 20 feet long, 20 feet wide, and 4 feet thick; each of these we will call B. Three smaller blocks, each 20 feet long, 4 feet wide, and 4 feet thick; each of these we will call C. |