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I. In the plane triangle ABC,
AC=104

ZA=270.59'
Given BC=70

Ans. Lo=1070.49'
LB=449.12

AB=142.02
Required the other parts.
2. In the plane triangle ABC,

SA
AC=104

S

Lc=720.11' or 1070.49* Given AB=142.02

Ans. {BC=13477 or 70
ZB=449.12

ambiguous. Required the other parts.

3. A man travels from A to B, 3•7 miles, then turning a little to the left hand goes from B to c, which is 4•7 miles : at che observes that A and B make an angle of 29o.16'. What is the distance from A by the shortest cut?

Answer, 7 miles.

4. A man travels from A to B, 3•7 miles; returning in a mist, he loses his way, and going a little too much towards the left hand, comes to c, which is 4.7 miles from B. It now clearing up he could see both A and B, and observes that they make an angle of 29:16!. Pray how far is he from A?

& the shor

(N) CASE III. Given two sides and the angle contained between them, to find the rest. The side AB=98

Required the angles a Given The side BC=95.12

and c, and the side ac. The angle B=114o.24'

BY CONSTRUCTION. Make AB=98 by a scale of equal parts, and the angle b=1140.24' by a scale of chords (R.27.); draw BC, which make equal to 95·12, and abc is the triangle required. The angles A and c will measure 329.15' and 33o.21', and AC will be 162.

A

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Or thus,

BY CALCULATION. See Rule II.
To find the angles.
AB + BC=193•12
2.28583 AB + BC=193.12

2.28583 : AB-BC=2.88 0.45939 : AB - BC=2.88

0.45939 : : co-tangent B = = 570.12' 9.80919 : : tang. supp. B=320.48' 9.80919 : tangent (ANC)=00. 33' 7.98275 : tangent (ANC)=00.33' 798275 $B=570 12' its comp. 320.48'

sum 33.21=LC sum 33.21=LC

diff.3215= LA. diff. 32.15= LA.

To find the side ac. sine Lc=330.21'

9.74017
: AB=98

1.99123
: : sine B=
=1140.24', or sine 650.96

9.95397
; sine AC=162.34

2.21043

(O) BY GUNTER'S SCALE. 1. Extend the compasses from 193•12 to 2.88 on the line of numbers, that extent will reach from 320.48' to 0°.33' on the line of tangents. This is the method of working such examples as this, but so small an angle as 33' is not contained on the scale.

2. Extend from 330.211 to 650.36' on the line of sines, that extent will reach from 98 to 162 on the line of numbers.

PRACTICAL EXAMPLES. 1. In the plane triangle ABC, AB= 345

LB=270.4'
Given 3 AC=174.07

Lo=1159.36
LA=370.20

BC-232
Required the other parts.

Ans. S<B=270.4

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2. In the plane triangle ABC,

S
AB=103

LB=720.20
Given ac=126

Ans. Lo=519.10
La=56*.30'

BC=110:3
Required the other parts.
(P) CASE IV. Given the three sides, to find the angles.

The side AB=98 Given The side BC=95:12 Required theangles A,B,andc.

The side ac=162:34

BY CONSTRUCTION. Draw the longest side ac=

B. 162.34 from a scale of equal parts; with AB=98 in your compasses (taken from the same scale) and one foot in a describe an arc; with BC=95.12 in

ED your compasses cross it in B; then abc is the triangle required. The angles measured by a scale of chords (S. 27.) will be A-32°.15', B=114o.34', and c-33o.21'. ,

BY CALCULATION. See Rule III. Let e be the middle of the base ac, and DB perpendicular to AC.

To find the segments AD and Dc of the base. double Ac=324.68

2.51145 : AB + BC = 199.12

2.28583 :: AB-BC = 2.88

0.45939 to ED=1•713

0.23377
Then į ac+ED=81•17 +1713=82•883 = AD, the greater segment, and } 10-
ID=81•17-1713=79.457 =DC, the less segment.
To find the angles in the right angled triangles adc and CDB.

1.92123
BC = 95.12

1.97827 : radius, sine 90° 10.00000 : radius, sine of 90°

1000000 ::AD=82-883 1.91846 ::CD=79.457

1.90019 : sine DBA = 570.45'

: sine DBC =- 560.39
or co-sine DAB=
=320.15'

9.92723
or co-sine DCB=339.21'

9.92186 Then DBA + DBC = 570.45' + 56°.39ʻ=114°.24' < ABC,

AB=98

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(Q) BY GUNTER'S SCALE, according to the first method. 1. Extend the compasses from 324•68 to 193.12 on the line of numbers, that extent will reach from 2:88 to 1.713 the distance of a perpendicular from the middle of the base.

2. Extend from 98 to 82.883 on the line of numbers, that extent will reach from 90° to 570.45' on the line of sines.

3. Extend from 95012 to 79.457 on the line of numbers, that extent will reach from 90° to 56o.39on the line of sines.

BY GUNTER'S SCALE, according to the second method. 1. Extend the compasses from half the sum of the three sides 177.73 to one of the containing sides AB=98, that extent will reach from ac=162.34 the other containing side, to a fourth number 89.5, on the line of numbers.

2. Extend the compasses from this fourth number 89.5 to (the difference between the half sum of the three sides and the side opposite to the angles sought), 82.61 on the line of numbers, that extent will reach from 90° on the line of sines to the required angle 32°, on the line of versed sines, immediately under the line of sines.

This is derived from the proportions in the investigation of Gunter's Rule (X. 22, and note).

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PRACTICAL EXAMPLES. 1. In the plane triangle ABC, AB= 142.02

LA=270.59
Given BC=70

Ans. LB=44°.12
AC=104

Lo=1070.49
Required the angles.
2. In the plane triangle ABC,

AB=112.65
Given BC=112

Ans. LB=46°.34'
AC=120

Lo=570.59'

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SCHOLIUM.

(R) There are some authors and teachers of trigonometry, who make no distinction of cases between right and obliqueangled triangles, but divide the whole into three cases; because the three rules necessary for solving the problems that occur in oblique trigonometry, are sufficient for solving those which occur in right-angled trigonometry. For instance, Rule I. (E. 52.) will solve all the cases in right-angled triangles (except the 6th), and the first and second case in obliqueangled triangles: Rule II. (G. 53.) will solve the 6th case in right-angled triangles and the 3d case in oblique; and Rule III. (H. 54.) will solve the last case in oblique triangles.

CHAP. III.

THE APPLICATION OF PLANE TRIGONOMETRY TO THE MEN

SURATION OF DISTANCES, HEIGHTS, &c.

(S) The mensuration of distances and heights depends upon the rules of plane trigonometry already explained, together with the use of certain instruments for taking angles.

(T) Horizontal and vertical angles are usually measured with a theodolite furnished with one or two telescopes, and a vertical arc; and if the horizontal and vertical arcs of the instrument be described with a radius of not less than 34 inches, the observed angles may be measured to half a minute, or the 120th part of a degree.

(U) Angles which are oblique to the horizon are generally taken with a sextant; which must be held in such a position, that its plane may coincide with the two objects and the eye. When vertical angles are taken with this instrument, an artificial horizon must be used, and the reflected image of the object from the glasses of the sextant must be brought to coincide with the reflected image of the same object in the artificial horizon.

(V) Base lines are generally measured with rods, or the four pole Gunter's chain; but common tape of 50 or 100 feet in length is often preferred both for accuracy and expedition: especially if it be kept dry, and the ground be tolerably level.

(W) The use of instruments must be acquired under the direction of a person well skilled in their several adjustments, as

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