Ex. 2. In a triangle ABC, there are given AB, 532, BC, 358, and the angle C, 107° 40', to find the other parts.

Ans. A 39° 52′ 52"; B=32° 27' 8"; AC-299.6. In this example there is no ambiguity, because the given angle is obtuse.

CASE III.

59. Given two sides and the included angle, to find the third side and the remaining angles.

The sum of the required angles. is found by subtracting the given angle from 180°. The difference of the required angles is then found by Theorem II. Half the difference added to half the sum gives the greater angle, and, subtracted, gives the less angle. The third side is then found by Theorem I.

Ex. 1. In the triangle ABC, the angle A is given 53° 8'; the side c, 420, and the side b, 535, to find the remaining parts. The sum of the angles B+C=180°-53° 8'=126° 52'. Half their sum is 63° 26'.

Then, by Theorem II.,

535+420:535–420 :: tang. 63° 26': tang. 13° 32′ 25′′, which is half the difference of the two required angles. Hence the angle B is 76° 58′ 25′′, and the angle C, 49° 53′ 35′′. To find the side a;.

sin. C:c: sin. A: a=439.32.

Ex. 2. Given the side c, 176, a, 133, and the included angle B, .73°, to find the remaining parts.

Ans. b=187.022, the angle C, 64° 9' 3", and A, 42° 50′ 57′′.

CASE IV.

60. Given the three sides, to find the angles,

Let fall a perpendicular upon the longest side from the opposite angle, dividing the given triangle into two right-angled triangles. The two segments of the base may be found by Theorem III. There will then be given the hypothenuse and one side of a right-angled triangle to find the angles.

Ex. 1. In the triangle ABC, the side a is 261, the side b, 345, and c, 395. What are the angles?

or

Let fall the perpendicular CD upon AB.
Then, by Theorem III.,

AB: AC+CB:: AC-CB: AD-DB;

395: 606 :: 84:128.87.

Half the difference of the segments added to half their sum gives the greater segment, and subtracted gives the less seg

b

A

D

ment.

Therefore AD is 261.935, and BD, 133.065.Then, in each of the right-angled triangles ACD, BCD we have given the hypothenuse and base, to find the angles by Case II. of Bright-angled triangles. Hence

AC: R::AD: cos. A=40° 36′ 13′′;

BC:R::BD: cos. B-59° 20′ 52".

Therefore the angle C=80° 2′ 55′′.

Ex. 2. If the three sides of a triangle are 150, 140, and 130, what are the angles?

"Ans. 67° 22′ 48′′, 59° 29′ 23′′, and 53° 7′ 49′′.

Examples for Practice.

1. Given two sides of a triangle, 478 and 567, and the included angle, 47° 30', to find the remaining parts.

2. Given the angle A, 56° 34', the opposite side, a, 735, and the side b, 576, to find the remaining parts.

3. Given the angle A, 65° 40', the angle B, 74° 20', and the side a, 275, to find the remaining parts.

4. Given the three sides, 742, 657, and 379, to find the angles.

5. Given the angle A, 116° 32', the opposite side, a, 492, and the. side c, 295, to find the remaining parts.

6. Given the angle C, 56° 18', the opposite side, c, 184, and the side b, 219, to find the remaining parts..

This problem admits of two answers.

7. Given the angle B; 68° 35' 27", the angle C, 44° 48′ 47′′, and the side c, 479, to find A, a, and b.

8. Given the angle A, 67° 23′ 56′′, the side a, 1486,73, and the side b, 2073.22, to find B, C, and c.

9. Given the angle C, 66° 3' 27", the side a, 897, and the side b, 571, to find A, B, and c.

10. Given a 2251, b=738, and c=830, to find A, B, and C.

INSTRUMENTS USED IN DRAWING.

61. The following are some of the most important instruments used in drawing.

I. The dividers consist of two legs, revolving upon a pivot at one extremity. The joints should be composed of two different metals, of unequal hardness: one part, for example, of steel, and the other of brass or sil

ver, in order that they may move upon each other with greater freedom. The points should be of

tempered steel, and, when the dividers are closed, they should meet with great exactness. The dividers are often furnished with various appendages, which are exceedingly convenient in drawing. Sometimes one of the legs is furnished with an adjusting screw, by which a slow motion may be given to one of the points, in which case they are called hair compasses. It is also useful to have a movable leg, which may be removed at pleasure, and other parts fitted to its place; as, for example, a long beam for drawing large circles, a pencil-point for drawing circles with a pencil,. an ink-point for drawing black circles, etc.

62. II. The parallel rule consists of two flat rules, made of wood or ivory, and connected together by two cross-bars of equal length, and parallel to each other. This instrument is useful for drawing a line parallel to a given line, through a given point.

For this purpose, place the edge of one of the flat rules against the given line, and move the other rule until its edge coincides with the given point. A line drawn along its edge will be parallel to the given line.

63. III. The protractor is used to lay down or to measure angles. It consists of a semicircle, usually of brass, and is divided into degrees, and sometimes smaller portions, the centre of the circle being indicated by a small notch.

To lay down an angle with the protractor, draw a base line,

60 70 80 90 80

70 60

and apply to it the edge of the protractor, so that its centre shall fall at the angular point. Count the degrees contained in the proposed angle on the limb of the circle, and mark the extremity of the arc with a fine dot. Re

move the instrument, and through the dot draw a line to the angular point; it will give the angle required. In a similar manner, the inclination of any two lines may be measured with the protractor.

64. IV. The plane scale is a ruler, frequently two feet in length, containing a line of equal parts, chords, sines, tangents, etc. For a scale of equal parts, a line is divided into inches and tenths of an inch, or half inches and twentieths. When smaller fractions are required, they are obtained by means of the diagonal scale, which is constructed in the following manner. Describe a square

inch, ABCD, and divide each of its sides into ten equal parts.

Draw diagonal lines from the first point of division on the upper line to the second on the lower; from the second on the upper line to the third on the lower, and so on. Draw, also, other lines parallel to AB, through the points of division of BC. Then, in the triangle ADE, the base, DE, is one tenth of an inch; and, since the line AD is divided into ten equal parts, and through the points of division lines are drawn parallel to the base, forming nine smaller triangles, the base of the least is one tenth of DE, that is, .01 of an inch; the base of the second is .02 of an inch; the third, .03, and so on. Thus the diagonal scale furnishes us hundredths of an inch.

To take off from the scale a line of given length, as, for example, 4.45 inches, place one foot of the dividers at F, on the sixth horizontal line, and extend the other foot to G, the fifth diagonal line.

A half inch or less is frequently subdivided in the same manner,

65. A line of chords, commonly marked CHO., is found on most plane scales, and is useful in setting off angles. To form this line, describe a circle with any convenient radius, and divide the circumference into degrees. Let the length of the chords for every degree of the quadrant be determined and laid off on a scale: this is called a line of chords.

Since the chord of 60° is equal to radius, in order to lay down

an angle, we take from the scale the chord of 60°, and with this radius describe an arc of a circle. Then take from the scale the chord of the given angle, and set it off upon the former arc. Through these two points of division draw lines to the centre of the circle, and they will contain the required angle.

The line of sines, commonly marked SIN., exhibits the lengths of the sines to every degree of the quadrant, to the same radius as the line of chords. The line of tangents and the line of secants are constructed in the same manner. Since the sine of 90° is equal to radius, and the secant of 0° is the same, the graduation on the line of secants begins where the line of sines ends.

On the back side of the plane scale are often found lines representing the logarithms of numbers, sinės, tangents, etc. This is called Gunter's Scale.

66. V. The Sector is a very convenient instrument in drawing. It is generally made of ivo

ry or brass, and consists of two equal arms, movable about a pivot as a centre, having several scales drawn on the faces, some single, others double. The

single scales are like those upon a common Gunter's Scale. The double scales are those which proceed from the centre, each being laid twice on the same face of the instrument, viz., once ön each leg. The double scales are a scale of lines marked Lin., or L.; the scale of chords, sines, etc. On each arm of the sector there is a diagonal line, which diverges from the central point like the radius of a circle, and these diagonal lines are divided into equal parts.