15. What is the square root of 5764801 ? 16. What is the square root of 1048576? 17. What is the square root of 282475249?

Ans. 16807.

NOTE 3. In extracting the root of a decimal, put the first point over hundredths and point toward the right, and if the last period is not full, annex 0

18. What is the square root of .4096?
19. What is the square root of .0625 ?
20. What is the square root of 39.0625?
21. What is the square root of 6046.6176?
22. What is the square root of 5.6 ?

OPERATION.

5.60 (2.36 +

4

43) 160

129

466) 3 100
2796

472) 30400

Ans. .64.

Ans. 6.25.

Ans. 2.36+.

If there is a remainder after employing all the periods in the given example, the operation may be continued at pleasure by annexing successive periods of ciphers, decimally; there will, however, in such examples, always be a remainder; for the right-hand figure

of the dividend is a cipher, whereas the right-hand figure of the subtrahend is, necessarily, the right-hand figure of the square of some one of the nine significant figures, the right-hand figure of the root and of the divisor being always alike. Now, no one of these nine figures, squared, will give a number ending with a cipher; ., the last figure of the dividend and of the subtrahend being unlike, there must be a remainder.

23. What is the square root of 2?

Ans. 1.41421+.

358. Explain Ex. 14. What is Note 3? Explain Ex. 22.

24. What is the square root of 20? 25. What is the square root of 316? 26. What is the square root of 31.6?

Ans. 4.472+

359. To extract the root of a common fraction, or of a mixed number:

RULE. Reduce the fraction or mixed number to its simplest form, and then take the root of the numerator and denominator separately; or, if either term of the fraction, when reduced, is an imperfect square, reduce the fraction to a decimal (Art. 173), and then proceed as in the foregoing examples.

27. What is the square root of 23?

360. A TRIANGLE is a figure bounded by three straight lines.

A right-angled triangle has one of its angles a right angle, as at C.

The side opposite the right angle is called the hypothenuse; the other two sides are the base and perpendicular.

359. Rule for extracting the root of a common fraction or mixed number? 360. What is a Triangle? A right-angled triangle? Hypothenuse? Base?

The square described on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides. Also the square of either of the two sides which form the right angle is equal to the square of the hypothenuse diminished by the square of the other side. This will be seen by counting the small squares in the square of the hypothenuse and

those in the squares of the other two sides, Hence,

1st. To find the hypothenuse when the base and perpendicular are given,

RULE. Add the square of the base to the square of the perpendicular, and extract the square root of the sum.

2d. To find either side about the right angle when the hypothenuse and the other side are given,

RULE. From the square of the hypothenuse, subtract the square of the other given side, and extract the square root of the remainder.

Ex. 1. The base of a right-angled triangle is 6 feet and the perpendicular is 8 feet; what is the hypothenuse?

=

6236, 8264; 36+64 100; /100=10. Ans. 10 ft. 2. The hypothenuse of a right-angled triangle is 15 and the base is 12; what is the perpendicular?

152=225, 122=144; 225- ∙144=81; 81=9, Ans.

360. The square of the hypothenuse equals what? The square of one of the other sides? How may this appear? Rule for finding the hypothenuse? Base or Perpendicu lar? Explain Ex. 1.

3. A ladder resting upon the ground 21 feet from a house, just reaches a window which is 28 feet high; how long is the ladder?

4. A tree that was 64 feet high is broken off 24 feet high, the part broken off turning upon the stub as upon a hinge; at what distance from the bottom of the tree does the top strike the ground? Ans. 32 ft.

5. Two vessels sail from the same port, one due east 40 miles and the other due south 9 miles; how far apart are they?

6. A general has 9801 men; if he places them in a square, how many will there be in rank and file?

7. How many rods of fence will be required to inclose 640 acres of land in a square form? Ans. 1280.

8. A farmer sets out an orchard of 600 trees so that the number of rows is to the number of trees in a row as 2 to 3. The trees are 25 feet apart and no tree is within 12 feet of the fence; how many square feet of land in the field?

D

A

Fig. 2.

B

361. In figure 3 we have combined a circle (Art. 109), a square (Art. 101, Note), and two equal right-angled triangles. The line AC is the diameter of the circle, the diagonal of the square and the hypothenuse of each of the triangles. The square is said to be inscribed in the circle and the circle is circumscribed about the square.

The diameter of any circle is to its circumference in the ratio of 1 to 3.141592, nearly; hence the diameter multiplied by 3.141592 will give the circumference, and the circumference divided by 3.141592 will give the diameter.

The area of a circle may be found by multiplying the square of its diameter by .785398, nearly, and if the area is divided by .785398, the quotient will be the square of the diameter.

361. What does Fig. 3 represent? What is the line AC? What is said of the square? Of the circle? Ratio of diameter to circumference? How is circumference found when diameter is given? Diameter when circumference is given? Area of a circle, how found! Diameter, when area is given?

362. Similar figures are figures that are of precisely the ame form, whether large or small.

The areas of all similar figures are to each other as the squares of their corresponding lines.

9. What is the diameter of a circular pond which shall contain 25 times as much area as one 8 rods in diameter? Ans. 40 rd. 10. The area of a triangle is 24 square inches and one side of it is 8 inches; what is the corresponding side of a similar triangle containing 96 square inches? Ans. 16 in.

11. What is the side of a square that shall contain 36 times as much area as one whose side is 5 feet?

12. What is the side of a square equal in area to a circle 100 feet in diameter ? Ans. 88.622. ft.

13. A circular field contains 10 acres; what is the length of its diameter?

14. What is the difference in the expense of fencing a circular 10-acre lot and one of the same area in a square form, the fence costing 75 c. per rod? Ans. $13.653.

15. If a pipe 3 inches in diameter will empty a cistern in 8 minutes, what is the diameter of the pipe which will empty it in 18 minutes?

16. The area of a rectangular piece of land (Art. 101, Note) is 50 acres, and the length of the piece is to its breadth as 5 to 1; what are the length and breadth?

17. A room is 16 ft. long, 12 ft. wide, and 9 ft. high; what is the distance from one lower corner to the opposite upper corner? Ans. 21.931

ft.

18. The diameter of a circle is 10 inches; how many inches in the side of the inscribed square? Ans. 507.071+. SOLUTION. By figure 3 it is seen that the diameter of the circle is the hypothenuse of a right-angled triangle whose other sides are equal to each other; .. the square of either side of the inscribed square is one half of the square of the diameter.

19. What is the side of the greatest square stick of timber that can be hewn from a log 18 inches in diameter?

362. What are similar figures? The ratio of the areas of similar figures?