PROOF. This work may now be proved by adding together all the square rods contained in the several parts of the figure, thus : A contains 30 x 30 = 900 square rods. B 30 x 6= 180 Or, by Involution, 6 x 6 = 36 36 X 36 = 1296, Ans., as before. 30 x 6 = 180 1296 square rods. From these illustrations we derive the following RULE. 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 oʻin 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. More Exercises for the Slate. 16. What is the square root of 65536 ? OPERATION. PROOF 2) 65536 ( 256, Ans. 256 4 256 1605 ) 80,25 80,25 4025 6440 0000 6480,25 18. What is the square root of 470596 ? A. 686. In this example, we have a remainder, after obtaining one figure in the roots In such cases, we may continue the operatiou 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 squaro 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, eveçao 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 fož almost any purpose. A. 6,3245 + 25. What is the square root of 30 ? A. 5,4772. 26. What is the square root of 104? A. =$. Or, we may reduce the given fraction to its lowest terms before the root is extracted. Thus, w 1944=nr , Ans., as before. 27. What is the square root of us? A. H. -28. What is the square root of ? A. 29. What is the square root of T23432T? A. TÓT: If the fraction be a surd, the easiest method of proceeding will be to redoco it to a decimal first, and extract its root afterwards. 30. What is the square root of 7%? A. 99128 +. 32. What is tho square root of 11 ? A. ,9574 +. 32. What is the square root of 1? A. ,83205. 33. What is the square root of 4203 ? 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. 203. 34. What is the square root of 91225? A. 303. 35. A general has an army of 5625 inen; how many must he place in rank and filo, to form them into a square? 5625 = 75, sins. 36. A square pavomenť contains 24336 square stones of equal size; how many are contained in one of its sides ? A. 156. 37. In a circle, whose aren, 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 X 4=20, 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. Brods. 41. If one side of a square picce of land measure 5 rods, what will the side of one measure, which is four times as large : 16 times as large? 36 times as large? A. 10. 20. 30. 42. A man is desirous of forming a tract of land, containing 140 acres, soods 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 tho square rods contained in said road? A. 240 rods. EXTRACTION OF THE CUBE ROOT. I LXXXVII. Q. Involution, (TLXXXIV.,) 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 (T LXXXVII.) was defined to be the extracting the 1st power vor roots of higher powers; can you tell me, then, what is tho cubo root of 277 A. 3. A. Because 3x3x, 3, or, expressed thus, 38=27. Ă. It is only to find that number, which, being multiplied into itself three times, will produce the given number. Q. We have seen, (F 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 fect, inches, &c., us are expressed by their product; but what do you suppose the shape of a solid body is, which is en 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. = 64. Q. Why? A. Because 38 = 27. Q. What is the length of each side of a cubical block containing 64 cubio inches ? A. 4 inches. Q. Why? A. Because 4 X 4 X4, or 48 = 64 cubic inches. Q. What is the length of each side of a cubical block containing 1000 cubio fcet ? A. 10. Q. Why? A. Because 103 =1000. 1. In a square box which will contain 1000 marbles, how many will it tako 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? A. 24. 3. What is the difference between 3/8 and 23 ? A. 6. 4. What is the difference between 3/1 and 13? A. 0. 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. O. 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 sido 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 op eration 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 tiine 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 comprehenlod 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, cach 20 feet long, 20 feet wide, and 4 feet thick; cach 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. One block, and the smallest, 4 feet long, 4 fect wide, and 4 feet thick ; this We will call D. We are now prepared to solve the preceding example. In this example, you recollect, we were to find the length of one side of the cubo, containing 13824 cubic feet. OPERATION 1st. In this example, we know that one side cannot be 30 feet, ft. 13824 ( 20, root. 203 = for 30o =27000 solid feet, being 8000 inore than 13824, the given 2 X 2 X 300 = divisor, 1200 ) 5824 ( 4 sum; therefore, we will take 20 for the length of one side of quot. 4 the cube. 4800 Then, 20 x 20 x 20 = 8000 2 X 30 X 4 X 4= 960 solid feet, which we must, of 4 X4 X4=64 course, deduct from 13824, leaving 5824. (See Operation 5624 deducted. Ist.) These 8690 solid feet, the pupil will perceive, are the 0000 solid contents of the cubical Or, block marked A. This corThe same operation, by neglecting the ciphers, responds with the operation; may be performed thus : for we write 20 feet, the length of the cube A, at the right of OPERATION 20. 13824, in the form of a quo13824 (20+4, or 24, root, tient; and its square, 8000, 8 Ans. under 13824; from which sub tracting 8000, leaves 5824, as 2X2 X 300 = 1200) 5824, dividend. before. As we have 5824 cubic feet 4800 2 X 30 X 4X4= remaining, we find the sides of 960 the cube A are not so long as 4 X 4X4= 64 they ought to be; consequently 5824, subtrahend. we must enlarge A; but in doing this, we must enlarge 0000 the three sides of A, in order that we may preserve tho cubical form of the block. We will now place the three blocks, each of which is marked B, on the three sides of A. Each of these blocks, in order to fit, must be as long and as wide as A; and, by examining then, you will see that this is the case ; that is, 20 feet long and 20 feet wide; then 20 x 20 = 400, the square contents in one B; and 3 x 400 = 1200, square contents in 3 Bs; then it is plain, that 5824 solid contents, divided by 1200, the square contents, will give the thickness of each block. But an easier method is, to square the 2, (tens,) in the root 20, making 4, and multiply the product, 4, by 300, making 1200, a divisor, the same as before. We do the same in the operation (which see); that is, we multiply the square of the quotient figure, 2, by 300, thus, 2 X 2=4 X 300 = 1200; then the divisor, 1200 (the square contents) is contained in 5824 (solid contents) 4 times; that is, 4 feet is the thickness of each block marked B. This quotient figure, 4, we place at the right of 5824, and then, 1200 square feet x 4 feet, the thickness, =4800 solid feet. If we now examine the block, thus increased by the addition of the 3 Bs, we shall see that there are yet three corners not filled up: these are reprosented by the three blocks, each marked C, and each of which, you will perceive, is as long as either of the Bs, that is, 20 feet, being the length of A, which is the 20 in the quotient. Their thickness and breadth are the same as the thickness of the Bs, which we found, by dividing, to be 4 feet, the last quotient figure. Now, to get the solid contents of each of these Cs, we multiply their thickness (4 feet) by their breadth (4 feet), = 16 square feet; that is, iho square of the last quotient figure, 4, = lò; these 16 square contents must be |