gure of the root. To obtain a fourth figure in the root, form another eriod by annexing two zeros, double 643, and so continue. From the above, we deduce the following RULE FOR EXTRACTING THE SQUARE ROOT OF A NUMBER. Point off the given number into periods of two figures each, y placing a dot over the units' figure and every alternate figure o the left in whole numbers, and to the right in decimals. Find the greatest square number in the left hand period, and write its root as the first term in the answer. Subtract the square umber from the left hand period, and to the remainder bring lown the next period for a dividend. Take twice the root already found for a trial divisor; rejecting he right hand figure of the dividend, divide it by the trial divisor; lace the result, as the second term in the root, also at the right of he trial divisor, making a true divisor; multiply the true divisor hus obtained by the last term of the root, and subtract this product from the dividend; to the remainder bring down the next period for a new dividend. Double the terms of the root already found for a new trial livisor, and proceed as before. NOTE I. - When a zero occurs in the root, annex a zero to the trial livisor, bring down another period, and proceed as before. NOTE II. - If a root figure proves too large, substitute a smaller, and epeat the work. NOTE III. - When a remainder occurs after all the periods are brought lown, the root may be more nearly approximated by annexing periods of eros, and continuing the operation. NOTE IV. The square root of a common fraction may be obtained by xtracting the root of both terms when they are perfect squares; when hey are not, the fraction may first be reduced to a decimal. NOTE V.Mixed numbers may be reduced to the decimal form, or to mproper fractions when the denominator of the fractional part is a square Lumber. 390. The above rule may be illustrated by diagrams. Fig. 1. 3600 sq. ft. 60 ft. Fig. 2. additions we wish to ascertain. 60 ft. 60 ft. M 60 ft. 0 If the 496 square feet equalled the feet in the side additions, M and N, the width of the additions would be determined by dividing 496 by twice the length of the square already found, 2X 60. Using this as the trial divisor, we obtain 4 as the width, which is the units' term of the root; but the entire length of the additions is two times the tens, plus the units, or 124 (Fig. 3), the product of which by 4, the units' term, is 496. There being no remainder, 4096 is found to be a square of which 64 is the root, and the length of the Fig. 3. 60 + + 4 court is 64 feet. 60 Let A B C D (Fig. 1) represent a square court containing 4096 square feet, the length of whose side we wish to determine. Having found (Art. 389) that the greatest square of tens in 4096 is 3600, the root of which is 6 tens, we deduct 3600 from 4096, and have left 496 square feet, which are to be disposed on two sides of the square already found. The width of these By extending the lines a and b, we shall divide the addition into three parts, M, N, and O; M and N having for one side the tens of the root, and O being a square whose side is equal to the width of the side additions. 391. 1. What is the square root of 841 ? EXAMPLES IN SQUARE ROOT. Ans. 29. Ans. 874. Ans. 3690. Ans. 503. Ans. 1006. Ans. 21.15. Ans. .075. Ans. .41109+. Ans. T 10. What is the square root of 17? NOTE.-. 11. What is the square root of 10? NOTE. - Extract the root in the following to five places. 16. √2 × (93)2=? 17. √21025? 31. √of of 1=? 32. √.144=? 33. √8=? 34.81.10083136=? 18. √980100 =? 19. √502681=? 20. √2216 = ? 21. 14002564? Ans. f. Ans. 31. Ans. .86602+. Ans. 2.8635+. Ans. 3.02334+. 35. √.7=? 36. √746841.64 =? 39. √.42025 =? 40. √ of .052 = ? 42. √√(.25÷.061)× (})2=? 392. PRACTICAL EXAMPLES. 1. There is a field of corn having an equal number of rows and hills in a row, which contains 1020100 hills in all; what is Ans. 1010 rows. the number of rows in the field? 2. A body of troops, consisting of 2601 men, has an equal number in rank and file; how many are there in each? Ans. 51 men. 3. A company of persons spent $3.24; each person spending as many cents as there were persons, how many cents did each Ans 18 cents. spend? 4. What is the length of one side of a square farm containing 302 acres, 2 roods of land? Ans. 220 rods. 5. What is the length of a square park which contains 2 square miles? Ans. 1.4142+ miles. 6. There is a circular lot which contains 3 acres; what is the length of a square lot whose area is the same? Ans. 21.9089+ rods. 7. What is the size of a square lot whose area is thirty times that of the above? Ans. 120 rods. 8. What is the cost of fencing a square lot which contains 1 acre, at $5 per rod? Ans. $252.98. 9. The side of a square is 8 ft. 6 in.; what is the side of a square having 25 times the area? 10. A owned a of land 51 rods by 80 rods, and another 180 rods by 100 rods, which he bartered with B for a square lot containing 138 acres; how many rods less of fencing are there in the square lot than in the other two? Ans. 228 rods nearly. 11. I have two square lots of land, the larger of which contains 270 acres; the ratio of the smaller to the larger is as 5 to 6; what is the length of one side of the smaller? Ans. 189.73+ rods. 12. On a roof there are laid 5000 slates,—the number in the length being twice the number in the breadth; what is the number each way? NOTE. It is evident that the slates are laid in two equal squares; hence the square root of of 5000 (V of 5000) will equal the breadth. Ans. 50 slates in breadth; 100 slates in length. 13. Suppose the above roof to have had 10000 slates, and the breadth to have been one third of the length, what would have been the number of slates in the length and breadth? Ans. 173.205+length; 57.735+ breadth. 14. What is the difference between the fencing of a 34-acre lot, whether it be a square or a rectangular lot, twice as long as it is wide? Ans. 17.89 rods. 15. My orchard contains 5400 trees; the number of trees in width is to the number in length, as 2 to 3; what is the number each way? NOTE. of the trees will be a square, whose square root will be the number of trees in the width of the orchard. 16. Suppose, in the above orchard, the outer rows of trees to stand upon the boundary line, and all to stand 30 feet apart, what is the area covered by the orchard? Ans. 10811 acres. 17. There is a rectangular court paved with 1728 paving. stones 15 inches square; the length of the court is to the width as 4 to 3; what is the number of stones each way? 18. How many square feet in the superficial contents of the above court? 19. What is the side of a square that will contain as many square feet as a rectangle whose sides are 150 and 70 feet? 20. What is the mean proportional between 6 and 24? (Art. 373.) APPLICATION OF SQUARE ROOT TO RIGHT-ANGLED TRI- DEFINITIONS. 393. An Angle is the opening between two lines that meet each other. 394. A Right Angle is the angle formed by two lines that are perpendicular to each other. (Art. 191.) 395. A Triangle is a figure having three angles, and bounded by three straight lines. 396. A Right-angled Triangle is a triangle having one of its angles a right angle. 397. The Hypothenuse of a Right-angled Triangle is the side opposite the right angle. 398. The Base of a Right-angled Triangle is the side upon which it is supposed to stand. 399. The Perpendicular of a Right-angled Triangle is the side perpendicular to the base. |