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Fifth : Bring down the pair 16 to the right side of 263, makes 26316; double the four figures in the quotient and set their product for a divisor as at 2924 under k ; seek how many times 2924 in 26316, rejecting the right hand 6; say no times; set a cipher in the quotient and another on the right of 29:24, makes 29240.

Sixth : Bring down the pair 81 to the right side of 26316, makes 2631681 for a new dividend ; seek how many times 29240 are in the dividend, rejecting the right hand figure 1; say 9 times; set 9 in the quotient and at the right of the cipher in the divisor, makes 292409; multiply as in Long Division and set the product under 2631681.

Now because there were 3 pairs in the decimals we will point off three decimals in the root : This gives for the root, 146.209; which if multiplied by itself, will produce the number it was extracted from, 21377.071681

LESSON 3.
What is the square root of .75?

Ans. .866+
:.75,00,00(.866
64

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LESSON 4. If an army consist of 222784 men, how many will each side contain when they are placed in form of a solid square ?

Answer, 472. ✓ 22,27,84.(472

16

87). 627

609

942). 1884

1884

LESSON 5. How large a square room may I floor with 5625 feet of plank?

Ans. 75 feet square. ✓ 56,25(75 root or answer.

49

145).725

725

LESSON 6. An irregular field of corn contains 197136 hills; how many hills must be placed in a row when the whole are formed into a square field?

Ans. 444. ✓ 19,71,36(444 root or answer.

16

84). 371

336

884). 3536

3536

LESSON 7. How long is the diagonal line of a field, in form of a parallelogram, 10 chains one way and 5 the other?

Ans. 1118 links + or 11 ch. 18 links +

Note 1. A diagonal line is a line drawn from corner to corner, as C B, dividing this figure into two right angled triangles.

A

B

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2. A parallelogram is an oblong or long square, as A B CD.

3. A right angle or corner contains 90 degrees, and is in the shape of a carpenter's square.

4. An angle greater than 90 degrees, is an obtuse angle, that is, dull, blunt, not sharp pointed.

5. An angle less than 90 degrees, is an acute angle, that is, a corner sharp pointed.

6. The diagonal line aforesaid is also the hypothenuse to the triangle, C B D.

7. The sides of right angled triangles are called legs; the side C D is the longest leg, or base ; the line from D to B is the shortest leg or perpendicular.

Now let us find the length of the diagonal or dotted line, that is, the hypothenuse to the triangle C D B.

RULE.

Add the squares of the two legs, and extract the Square Root of their sum.

The leg CD

10 chains.
10

Square of C D

100
25 square of D B.

125 sum of the squares.

7 of D B 5= 25.

ch.
✓ 1,25.(11.18 + root or answer.
1

Proof

J: 11.18 21)025

11.18 21

8944 221)400*

1118 221

12298

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LESSON 8. The wall of a fort is 24 feet in height, and a ditch of 12 feet wide surrounds the fort; I wish to know the length of a ladder that will reach from the outer edge of the ditch to the top of the wall?

Ans. 26.83 + feet.
The longest leg 24x24=576
The short leg 12x12=144
Sum of the squares,

720 feet.
v 7,20.00,00,(26.83 + feet.
4

26.83

12
46)320
276

in. 9.96

12
528)4400
4224

11".52

12
5363)17600
16089

6".24

Remainder, 1511

In this case, the height of the wall is the longest leg, the distance from the bottom of the wall to the outer edge of the ditch the short leg, and the length of the ladder the hypothenuse.

LESSON 9. The hypothenuse and one leg or side of a triangle given, to find

the other side.

RULE. From the square of the hypothenuse, subtract the square of the given leg; the square root of the difference, is the length of the leg required. In the triangle A B C, the

A hypothenuse À B, 30.02, and the side BC 26, are given to find the perpendicular A C.

30.02 ch.

B

26 ch. The hypothenuse 3002 links. The base B.C, 26.00

3002 26.00

6004 156

900600 52

9012004 square. 6760000

6760000

Difference, v 2,25,20,04(1500 links, Ans.

1

25)125

125

3000)...2004 1500 links=15 chains the length of A C.

LESSON 10. A steeple 120 feet in height has a ladder of the same length placed with the foot 20 feet from the bottom of the steeple, on a level surface; how far does the top of the ladder come below the top of the steeple?

Answer, i foot 8 in. 1" 11" +

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