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Then the other parts being measured, they are found to be nearly as follows; viz. the side BC 232 yards, the angle B 27°, and the angle C 115o1⁄2,

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In the first proportion.-Extend the compasses from 519 to 171, on the line of numbers; and that extent will reach, on the tangents, from 71° (the contrary way, because the tangents are set back again from 45°), a little beyond 45, which being set so far back from 45, falls upon 44°4, the fourth term.

In the second proportion. - Extend from 64° to 37°, on the sines; and that extent will reach, on the numbers, from 345 to 232, the fourth term sought.

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THEOREM III.

When the three sides of the triangle are given.

Then, having let fall a perpendicular from the greatest angle upon the oppo site side, or base, dividing it into two segments, and the whole triangle into two right-angled triangles; it will be,

As the base, or sum of the segments,

Is to the sum of the other two sides;

So is the difference of those sides,

To the difference of the segments of the base.

Then half the difference of the segments being added to the half sum, or the half base, gives the greater segment; and the same subtracted gives the less segment.

Hence, in each of the two right-angled triangles, there will be known two sides, and the angle opposite to one of them; consequently, the other angles will be found by the first problem.

Demonstr.-By Cor. to Theorem 35, Geometry, the rectangle under the sum and difference of the two sides, is equal to the rectangle under the sum and difference of the two segments. Therefore, by forming the sides of these rectangles into a proportion, it will appear that the sums and differences are proportional as in this theorem, by Theor. 76, Geometry.

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centre A, describe an arc;

arc, cutting the former in C.

With radius 232, and

and with radius 174, and centre B, describe another

Join AC, BC, and it is done.

Then, by measuring the angles, they will be found to be nearly as follow; viz. angle A 27o, angle B 37°, and angle C 115o1⁄2.

2. Arithmetically.

Having let fall the perpendicular CP, it will be,

As the base AB: AC + BC :: AC - BC: AP

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that is, as 345:406-07 :: 57·93: 68·18 = AP

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In the first proportion.—Extend the compasses from 345 to 406, on the line of numbers; then that extent will reach, on the same line, from 58 to 68.2 nearly, which is the difference of the segments of the base.

In the second proportion.-Extend from 232 to 206, on the line of numbers; then that extent will reach, on the sines, from 90° to 63°.

In the third proportion. - Extend from 174 to 138; then that extent will

-

reach from 90° to 52° on the sines.

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The three foregoing theorems include all the cases of plane triangles, both right-angled and oblique; besides which, there are other theorems suited to some particular forms of triangles, which are sometimes more expeditious in their use than the general ones; one of which, as the case for which it serves so frequently occurs, may be here taken, as follows:

THEOREM IV.

When, in a right-angled triangle, there are given one leg and the angles; to find the other leg or the hypothenuse; it will be,

As radius, i. e. sine of 90° or tangent of 45°

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C

F

F

Demonstr. AB being the given leg, in the right-angled triangle ABC; with the centre A, and any assumed radius AD, describe an arc DE, and draw DF perpendicular to AB, or parallel to BC. Now it is evident, from the definitions, that DF is the tangent, and AF the secant, of the arc DE, or of the angle A which is measured by that are, to the radius AD. Then, because of the parallels BC, DF, it will be as AD: AB :: DF: BC :: AF: AC, which is the same as the theorem is in words.

A

D

B

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Make AB 162 equal parts, and the angle A = 53° 7′ 48′′; then raise the perpendicular BC, meeting AC in C. So shall AC measure 270, and BC 216.

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Extend the compasses from 45° to 53°, on the tangents. Then that extent will reach from 162 to 216 on the line of numbers.

EXAMPLE II.

In the right-angled triangle ABC,

Given {the angle A 62° 40′

the leg AB 180

To find the other two sides.

Ans. SAC 392-0147
BC 348 2464

Note. There is sometimes given another method for right-angled triangles, which is this:

ABC being such a triangle, make one leg AB radius, that is, with centre A, and distance AB, describe an arc BF. Then it is evident that the other leg BC represents the tangent, and the hypothenuse AC the secant, of the arc BF, or of the angle A.

In like manner, if the leg BC be made radius; then the other leg AB will represent the tangent, and the hypothenuse AC the secant, of the arc BG or angle C.

But if the hypothenuse be made radius; then each leg

F

B

E

will represent the sine of its opposite angle; namely, the leg AB the sine of the arc AE or angle C, and the leg BC the sine of the arc CD or angle A.

And then the general rule for all these cases, is this, namely, that the sides of the triangle bear to each other the same proportion as the parts which they represent.

And this is called, Making every side radius.

OF HEIGHTS AND DISTANCES, &c.

By the mensuration and protraction of lines and angles, are determined the lengths, heights, depths, and distances of bodies or objects.

Accessible lines are measured by applying to them some certain measure a number of times, as an inch, or foot, or yard. But inaccessible lines must be measured by taking angles, or by some such method, drawn from the principles of geometry.

When instruments are used for taking the magnitude of the angles in degrees, the lines are then calculated by trigonometry in the other methods, the lines are calculated from the principle of similar triangles, without regard to the measure of the angles.

Angles of elevation, or of depression, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet suspended from the centre, and two sides fixed on one of the radii, or else with telescopic sights.

To take an angle of altitude and depression with the quadrant. Let A be any object, as the sun, moon, or a star, or the top of a tower, or hill, or other eminence and let it be required to find the measure of the angle ABC, which a line drawn from the object makes with the horizontal line BC.

D

Fix the centre of the quadrant in the angular point, and move it round there as a centre, till with one eye at D, the other being shut, you perceive the object A through the sights: then will the arc GH of the quadrant, cut off by the plumb line BH, be the measure of the angle ABC as required.

The angle ABC of depression of any object A, is taken in the same manner; except that here the eye is applied to the centre, and the measure of the angle is the arc GH, on the other side of the plumb line.

G

H

G

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The following examples are to be constructed and calculated by the foregoing methods, treated of in Trigonometry.

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