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3027756 1009252 3027756 2018504 25231300

Square,

254647.399876 Common Multiplier,

.7854

1018589599504 1273236999380 2037179199008 1782531799132

200000.0678626104

LESSON 22. A and B in partnership bought a grindstone which contained 3 feet in diameter; after making an allowance for for the eye, 4 inches by 3, the parties agreed that A should grind off one half first; how many inches of the diameter must A wear off for his part?

Answer, 10 inches 5 seconds 7 thirds.

First consider we have a diameter given to find the area.
See rule 4, under note 2, lesson 16.
The diameter is

3
3

The

The eye containing 12 su

perficial inches, or 14 of a Square of the diameter, 9 foot must be reduced to the Common multiplier, 7.854 decimal of a foot and sub

tracted from the area which Area including the eye, 7.0686 ||will give a new area,

00180 La reduce 144)12.0000.0833

432
1152 [area of
- the eye.

.480
00480

432

Area including the eye,

Area of the eye,
New area excluding the eye,

. 48 7.0686 feet, .0833

6.9853

Here 6.9853 is an area given to find the diameter. When found, we will call it “ a new diameter." See Rule under Lesson 15. New Area. ✓ 6.98,53(2.64 Root of the new area.

4

46)298

Common multiplier, 1.12837
276
Root of new area,

2.64 524). 2253

451348 2096

677022

225674 . 157

New diameter, 2.9788968 Now we have a new diameter given to form another which will make a circle half as large in area for B’s part.

See Rule 2, above example 2, under Lesson 16.
- The new diameter, 2.9788968

2.9788968

238311744 178733808 268100712 238311744 238311744 208522776 268100712 59577936

8.87382614505024 This square of the new diameter must be divided by the proportion 2; that is done to make another circle half as large by extracting the Square Root of the quotient.

2)8.87382614505024
4.43691307252512 half the square.

Carried forward.
To prove this operation, find the area of B's part by
Rule 4, under Note 2, in Lesson 16.

B's part must be equal to half the new area after the eye was deducted.

✓ 4.43,69,13,07,25,25,12
4

Root, (2.1063981 or B's diame

ter, which subtracted from the 41)43

new diameter gives a remainder 41

for A's part. 4206)26913

25236

[blocks in formation]

Suppose a ship sail from New-York 4000 miles south, then east 6000 miles; how far will she be from New York on a direct course, and what will be the course back ?

Ans. the ship will be 7211 miles from New York, and the course back will be N. 56° 19' W.

See Rules under Lesson 7.

One leg of the triangle, 4000

4000

Square,

16000000
Square of the other leg, 36000000
Sum of the squares, v 52,00,00,00(7211 answer.

49

14:). 300

284

1441). 1600

1441

14421). 15900

14421

.1479

Rule to find the course. As the sum of the hypothenuse and half the longest leg,

is to 86 degrees; So is the short leg, to its opposite angle. The longest leg is

6000 miles, Half the longest leg is

3000 miles. The hypothenuse is

7211 The sum of the hypothenuse and half

10211
the longest leg is
Sum, &c.

Short leg.
10211 : 86 :: 4000

4000

alf}

10211)344000(33 degrees.

30633

37670
30633

.7037

60 minutes in a degree. 10211)422220(41 minutes.

[Continued. 10211)422220(41 minutes

40844

.13780
10211

..3569 33 degrees and 41 minutes is the angle sought.

Subtract this angle from 90, the remainder will be 56 degrees, 19 minutes, the course to steer back.

90°
33 41

56 19 Navigators calculate by Points: 114, 11.25, or 11 de. grees 15 minutes make 1 point, 32 points the whole circle or 360 degrees.

11.25 degrees.

32 points.

2250 3375

360.00 degrees.

LESSON 24. The sum of two numbers and their product given to find each number.

The sum of two numbers when added together is 61, and their product is 624; what are those two numbers ?

Answer, 13 and 48.

RULES. 1. Square their sum,

61x61=3721 2. Subtract 4 times their product,

624x4=2496 from their square. 3. Extract the square root of the remainder, v12,25(35

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65)325

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