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Rule. — 1. Separate 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, and these points will show the number of figures of which the root will consist.
2. Find the greatest square number in the first or left hand period, placing 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, subtracting 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. Find how often the divisor is contained in the dividend, omitting, 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 the 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 all the periods have been brought down.
NoTE. . - 1. If the dividend does not contain the divisor, a cipher must be placed in the root, and also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend.
2. When there is a remainder after extracting the root of a number, periods of ciphers may be annexed, and the figures of the root thus obtained will be decimals.
3. If the given number is a decimal, or a whole number and a decimal, the root is extracted in the same manner as in whole numbers, except in pointing off the decimals either alone or in connection with the whole number, we begin at the separatrix and place a point over every second figure toward the right, filling the last period, if incomplete, with a cipher.
4. The square root of any number ending with 2, 3, 7, or 8, cannot be exactly found.
EXAMPLES FOR PRACTICE. 2. What is the square root of 148996 ?
* The figure of the root must generally be diminished by one or two units, on account of the deficiency in enlarging the square.
QUESTIONS. – What is the rule for extracting the square root ? What is to be done if the dividend does not contain the divisor? What must be done if there is a remainder after extracting the root ? What do you do if the given number is a decimal ? Of what numbers can the square root not be found ?
3. What is the square root of 516961 ? Ans. 719. 4. What is the square root of 182329 ? Ans. 427. 5. What is the square root of 23804641 ? Ans. 4879. 6. What is the square root of 10673289 ? Ans. 3267. 7. What is the square root of 20894041 ? Ans. 4571. 8. What is the square root of 42025 ?
Ans. 205. 9. What is the square root of 1014049 ? Ans. 1007. 10. What is the square root of 538 ? Ans. 23.1947. 11. What is the square root of 71? Ans. 8.42612. What is the square root of 7? Ans. 2.645+. 13. What is the square root of .1024 ?
Ans. .32. 14. What is the square root of .3364?
Ans. .58. 15. What is the square root of .895 ? Ans. .946+: 16. What is the square root of .120409? Ans. .347. 17. What is the square root of 61723020.96 ? Ans. 7856.4. 18. What is the square root of 9754.60423716 ?
Ans. 98.7654. Art. 269. If it is required to extract the square root of a vulgar fraction, or of a mixed number, the mixed number must be reduced to an improper fraction; and in both cases the fractions must be reduced to their lowest terms, and the root of the numerator and denominator extracted.
Note. – When the exact root of the terms of a fraction cannot be found, it must be reduced to a decimal, and the root of the decimal extracted.
EXAMPLES FOR PRACTICE.
Ans. 67. 4. What is the square root of 129495? 5. What is the square root of 6076?
Ans. 74. 6. What is the square root of 2837 ?
QUESTIONS. — Art. 269. What do you do when it is required to extract the square root of a vulgar fraction or of a mixed number?
7. What is the square root of 4717?
Ans. 65. 8. What is the square root of ? ? Ans. .858+. 9. What is the square root of 83}? Ans. 9.147 10. What is the square root of 12113? Ans. 11.042+.
3394 11. What is the square root of ?
Ans. $. 462
7673 12. What is the square root of
APPLICATION OF THE SQUARE ROOT. Art. 270. The square root may be applied to finding the dimensions and areas of squares, triangles, circles, and other surfaces.
1. 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. 2. A gentleman purchased a lot of land in the form of a square, containing 640 acres ; how many rods square is his lot?
Ans. 320 rods. 3. I have three pieces of land; the first is 125 rods long, and 53 wide; the second is 62} rods long, and 34 wide ; and the third contains 37 acres; what will be the length of the side of a square field whose area will be equal to the three pieces ?
Ans. 121.11+ rods. 4. W. Scott has 2 house lots; the first is 242 feet square, and the second contains 9 times the area of the first ; how many feet square is the second ?
Ans. 726 feet. 5. There are two pastures, one of which contains 124 acres, and the area of the other is to the former as 5 to 4; how many rods square is the latter ?
Ans. 157.48+ rods. 6. I wish to set out an orchard containing 216 fruit trees, so that the length shall be to the breadth as 3 to 2, and the distance of the trees from each other 25 feet; how many trees will there be in a row each way, and how many square feet of ground will the orchard cover? Ans. 18 in length; 12 in breadth; 116875 sq. ft.
Art. 271. A TRIANGLE is a figure having three sides and three angles.
A right angled triangle is a figure having three sides and three angles, one of which is a right angle.
QUESTIONS. - Art. 270. To what may the square root be applied ? - Art. 271. What is a triangle? What is a right angled triangle ? " What is the longest side called? What the other two?
The side A B is called the base of the triangle, the side B C the perpendicular, the side A C the hypothenuse, and the angle at B is a right angle.
Art. 272. In every right angled triangle the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular, as shown by the following diagram.
It will be seen by examining this diagram that the large square, formed on the hypothenuse À C, contains the same number of small squares as the other two counted together. Hence, the propriety of the following rules.
Art. 273. To find the hypothenuse, the base and perpendicular being given.
Rule. — Add the square of the base to the square of the perpendicular, and extract the square root of their sum.
Art. 274. To find the perpendicular, the base and hypothenuse being given.
RULE. Subtract the square of the base from the square of the hypothenuse, and extract the square root of the remainder.
Art. 275. To find the base, the hypothenuse and perpendicular being given.
RULE. — Subtract the square of the perpendicular from the square of the hypothenuse, and extract the square root of the remainder.
EXAMPLES FOR PRACTICE. 1. 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.
Questions. — Art. 272. How does the square of the hypothenuse compare with the base and perpendicular? How does this fact appear from Fig. 2?Art. 273. What is the rule for finding the hypothenuse? --Art. 274. What for finding the perpendicular? – Art. 275. What for finding the base ?
2. 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. 3. The hypothenuse of a certain right angled triangle is 60 feet, and the perpendicular is 36 feet; what is the length of the base ?
Ans. 48 feet. 4. A line drawn from the top of the steeple of a certain meeting-house to a point at the distance of 50 feet on a level from the base of the steeple, is 120 feet in length; what is the height of the steeple ?
Ans. 109.08+ feet. 5. The height of a tree on an island in a certain river, is 160 feet. The base of the tree is 100 feet on a horizontal line from the river, and is elevated 20 feet above its surface. A line extending from the top of the tree to the further shore of the river is 500 feet. Required the width of the river.
Ans. 366.47+ feet. 6. On the summit of a hill there is a tower 160 feet high, whose base is 90 feet, on a level, from a certain road that is 110 feet wide; the length of a line extending from the top of the tower to a point on the opposite side of the road is 300 feet. What is the elevation of the base of the tower above the road?
Ans. 63.64+ feet. 7. John Snow's dwelling is 60 rods north of the meetinghouse, James Briggs's is 80 rods east of the meeting-house, Samuel Jenkins's is 70 rods south, and James Emerson's 90 rods west of the meeting-house; how far will Snow have to travel to visit his three neighbors, and then return home?
Ans. 428.4+ rods. 8. A certain room is 24 feet long, 18 feet wide, and 12 feet high ; required the distance from one of the luwer corners to an opposite upper corner.
Ans. 32.3+ feet.
Art. 276. A CIRCLE is a plane figure bounded by a curved line, every part of which is equally distant from a point called the centre.
The circumference or periphery of a circle is the line which bounds it.
The diameter of a circle is a line drawn through the centre, and terminated by the circumference; as A B.
QUESTIONS. — Art. 276. What is a circle? What is the circumference of a circle ? What the diameter ?