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Subtract, and their difference will be

23.77 the least segment DB, In the right angled triangle ADC, there is AC47, and AD 40. 23, given, to find the angle A.

This is resolved by case 4. of right angled plane trigonometry, thus,

AD:R::AC: Sec. A.
40. 23: 900 :: 47: 31° 08'

Or it may be had by finding the angle ACD, the complement of the angle A ; without a secant, thus,

AC:R::AD: S. ACD.
47 : 90° : : 40 23 : 58° 52

90 — 58° 52'=31°. 08, the angle A.

Then by theo. 1. of this sect.

BC: S. A:: AC: S. B.
34 : 31° 08':: 47: 45° 37.

By cor. 1. theo.5. sect. 4. 180° — the sum of A and B-C

A 310.08
B 45. 37

180°—76. 45=103. 15', the angle C:

3d. By Gunter's Scale.

The first proportion is extended on the line of numbers; and it is no matter whether you extend from the first to the third, or to the second term, since they are all of the same kind: If you extend to the second, that distance applied to the third, will give the fourth ; but if you extend from the first to the third, that extent will reach from the second to the fourth.

The methods of extending the other proportions have been already fully treated of.

An example in each case of oblique angled triangles.

AC 290 ) A
1. Given, C 69.30' B required.

AB 350 BC

C 24°. 20 AB 2. Given, B 128. 30 required.

AC 3246) BC

AC 6 A

{
3. Given, C 124°.30' B

B required.
BC 4.5° SAB

SAB

4. Given,

AC
BC

46
92
52

A
B required.

Additional Exercises with their Ansners.

QUESTIONS FOR EXERCISE.

1. Given the Hypothenuse 108 and the Angle opposite the Perpendicular 25° 36 ; required the Base and Perpendicular.

Answer. The Base is 97.4, and the Perpendicular 46.66.

2. Given the Base 96 and its opposite Angle 71 45'; required the Perpendicular and the Hypothenuse.

Answer. The Perpendicular is 31.66 and the Hypothenuse 101.1,

3. Given the Perpendicular 360 and its opposite Angle 58° 20'; required the Base and the Hypothenuse.

Answer. The Base is 222, and the Hypothenuse 423,

4. Given the Base 720 and the Hypothenuse 980 ; required the Angles and the Perpendicular.

Answer. The Angles are 47° 17'. and 42° 43, and the Perpendicular 664.8

5. Given the Perpendicular 110.3 and the Hypothenuse 176.5; required the Angles and the Base.

Answer. The Angles are 38° 41' and 51° 19, and the Base 137.8.

6. Given the Base 360 and the Perpendicular 180; required the Angles and the Hypothenuse.

Answer. The Angles are 53° 8' and 36° 52, and the Hypothenuse 600.

7. Given one Side 129, an adjacent Angle 56° 30, and the opposite Angle 81° 36' : required the third Angle and the remaining Sides.

Answer. The third Angle is 41° 54', and the remaining Sides are 108.7 and 87.08.

8. Given one Side 96.5, another Side 59.7, and the Angle opposite the latter Side 31° 30': required the remaining Angles and the third Side.

Answer. This Question is ambiguous; the given Side opposite the given Angle being less than the other given Side (see Rule I. ;) hence, if the Angle opposite the Side 96.5 be acute, it will be 57° 38', the remaining Angle 90° 52', and the third Side 114.2; but if the Angle opposite the Side 96.5 be obtuse, it will be 122° 22', the remaining Angle 26° 8', and the third Side 50.32.

9. Given one Side 110, another Side 102, and the contained Angle 113° 36': required the remaining Angles and the third Side.

Answer. The remaining Angles are 34° 37' and 31° 47', and the third Side is 177.5.

10. Given the three Sides respectively, 120.6, 125.5, and 146.7 : required the Angles.

Answer. The Angles are 51° 53', 54° 58', and 73° 9'.

The student, who has advanced thus far in this work with diligence and active curiosity, is now prepared to study, with ease and pleasure, the following part; which comprehends all the necessary directions for the practice of Surveying.

PART II,

Or the Practical Surveyor's Guide.

SECT. I.

Containing a particular Description of the several Instruments

izsed in Surveying, with their respective Uses.

THE CHAIN.

THE stationary distance, or merings of ground, are measured either by Gunter's chain of four poles or perches, which consists of 100 links ; (and this is the most natural division) or by one of 50 links, which contains two poles or perches : but because the length of a perch differs in many places, therefore the length of chains and their respective links will differ also.

The English statute-perch is 54 yards, the twopole chain is 11 yards, and the four-pole one is 22 yards ; hence the length of a link in a statutechain is 7.92 inches.

There are other perches used in different parts of England, as the perch of noodland measure, which is 6 yards; that of church-land measure, which is 7 yards, and the forest measure perch, which is 8 yards.

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