OPERATION. have 625 tile, each of which is one foot square ; we wish to know the side of a square room, whose floor they will pave or cover. If we find a number multiplied into itself, that will produce 625, that number will be the side of a square room, which will require 625 tiles to cover its floor. We perceive that our number (625) consists of three figures, therefore, there will be two figures in the root ; for the product of any two numbers can have, at most, but just so many figures, as there are in both factors, and, at least, but one less. We will, therefore, for convenience, divide our number (625) into two parts, called periods, writing a point over the right hand figure of 625 (25 Ans. each period ; thus, 625. We 400 now find, that the greatest square 45) 2 25 number in the left hand period, 225 6 (hundred), is 4 (hundred); and that its root is 2, which we write in the quotient (see operation), As this 2 is in the place of tens, its value must be 20 and its square 400. Let this be represented by a square, whose sides measure 20 feet each, and whose contents will, therefore, be 400 square feet. (See figure 1.) D subtract 400 from 625, and there re 20 20 mains 225 square feet, to be arranged 20 on two sides of figure 1, in order that 400 its form may remain square. We therefore double the root 20, one of 20 the sides, and it gives the length of the two sides to be enlarged ; viz. 40. We then inquire, how many times 40, as a divisor, is contained in the dividend, and find it to be 5 times ; this we write in the root, and also in the divisor. This 5 is the breadth of the addition to our square. (See figure 2.). And this breadth, multiplied by the length of the two additions (40) gives the contents of the two figures, E and F, 200 square feet, which is 100 feet for each. There now remains the figure G, to complete the square, each side of which is 5 feet ; it being equal to FIG. I. 20 We now FIG. 11. 20 the breadth of the additions E E20 G 5 5 account of this last addition, that 100 25 the last figure of the root is placed in the divisor. If we now multiply the divisor, 45, by the last figure 20 D20 20 20 5 20 in the root (5), the product will 400 100 be 225, which is equal to the remaining feet, after we have form 20 ed our first square, and equal to the additions E, F, and G, in fig D contains 400 square feet. do. do. do. ure 2. We therefore perceive, F do. 100 do. do. that figure 2 may represent a G do. do. do. floor 25 feet square, containing Proof. 625 625 square feet. From the above, we infer the following 25 x 25 = 625. F 100 25 or, RULE. 1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points show the number of figures the root will consist of. 2. Find the greatest square number in the first or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the double of the root already found, on the left hand of the dividend for a divisor. 4. Seek how often the divisor is contained in the dividend, (except the right hand figure,) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor. Multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend. To the remainder join the next period for a new dividend. 5. Double the figures already found in the root for a new divisor, (or, bring down your last divisor for a new one, doubling the right hand figure of it,) and from these find the next figure in the root, as last directed, and continue the operation in the same manner, till you have brought down all the periods. Note 1. If, when the given power is pointed off, as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period. NOTE 2. If there be decimals in the given number, it must be pointed both ways from the place of units. If, when there are inte. gers, the first period in the decimals be deficient, it may be completed by annexing so many ciphers as the power requires. And the root must be made to consist of so many whole numbers and decimals, as there are periods belonging to each; and when the periods belonging to the given numbers are exhausted, the operation may be continued at pleasure by annexing ciphers. Note 3. If it be required to extract the square root of a vulgar fraction, reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and of the denominator for a new denominator ; or, reduce the vulgar fraction to a decimal, and extract its root. 2. What is the square root of 148996 ? OPERATION. 14 8996(386 9 68) 589 544 766) 4596 4596 3. What is the square root of 23804641 ? Ans. 4879. 4. What is the square root of 10673289 ? Ans. 3267. 5. What is the square root of 20894041 ? Ans. 4571. 6. What is the square root of 1014049 ? Ans. 1007. 7. What is the square root of 516961 ? Ans. 719. 8. What is the square root of 182329 ? Ans. 427. 9. What is the square root of 61723020.96 ? Ans. 7856.4. 10. What is the square root of 9754.60423716 ? Ans. 98.7654. 11. What is the square root of at ? Ans. 67: 12. What is the square root of 1848? 13. What is the square root of 6? 14. What is the square root of ? 15. What is the square root of 6016 ? 16. What is the square root of 2881 ? 17. What is the square root of 4747 ? Ans. 1 il Ans. 23. Ans. . Ans. 73 Ans. 53 Ans. 63. APPLICATION OF THE SQUARE ROOT. 18. A certain general has an army of 226576 men; how many must he place rank and file to form them into a square ? Ans. 476. Note. In a right angle triangle, the square of the longest side is equal to the sum of the squares of the other two sides. 19. What must be the length of a ladder to reach to the top of a house 40 feet in height ; the bottom of the ladder being placed 9 feet from the sill? Ans. 41 feet. 20. Two vessels sail from the same port; one sails due north 360 miles, and the other due east 450 miles ; what is their distance from each other ? Ans. 576.2+ miles. 21. If a pipe, 2 inches in diameter, will fill a cistern in 20 minutes, how long would it take a pipe, that is 3 inches in diameter ? Ans. 9 minutes. 22. If an anchor, which weighs 2000 lbs., requires a cable 3 inches in diameter, what should be the diameter of a cable, when the anchor weighs 4000lbs. ? Ans. 4.24+ inches. 23. How large a square stick may be hewn from a round one, which is 30 inches in diameter ? Ans. 21.2+ inches square. 24. John Snow's dwelling is 60 rods north of the meetinghouse, James Briggs' is 80 rods east of the meetinghouse, Samuel Jenkins' is 70 rods south, and James Emerson's 90 rods west of the meetinghouse ; how far will Snow have to travel to visit his three neighbours, and then return home ? Ans. 428.4+ rods. Section 49. EXTRACTION OF THE CUBE ROOT. A Cube is a solid, bounded by six equal squares. A number is said to be cubed, when it is multiplied into its square: To extract the cube root, is to find a number, which, being multiplied into its square, will produce the given number. The extraction of this root has been illustrated by mathematicians in various ways. But it is believed, that Robert Record, Esquire, of London, in his Arithmetic published in 1673, was among the first, who illustrated this rule by the use of various diagrams and blocks. The same thing, with but little variation, has been done by several arithmeticians in our own country, The Rule for extracting the root depends on the following THEOREM. If any line or number be divided into two parts, the cube of the whole line or number, is equal to the cube of the greater part, plus the square of the greater part multiplied by 3 times the less part, plus the square of the less part multiplied by 3 times the larger part, plus the cube of the less part. To illustrate this Theorem, let 27 be divided into two parts, 20 and 7. Then, by the hypothesis, the cube of 27 is equal to the cube of 20, plus the square of 20 multiplied by 3 times 7, plus the square of 7 multiplied by 3 times 20, plus the cube of 7. OPERATION. Cube of 27 =19683 8000 343 Proof. = 19683. |