9. If the three angles of a plane triangle be 106° 42′ 12′′, 46° 24′ 5′′, and 26° 53′ 43′′, and the side opposite the greatest angle 302.65 yds, what are the other two sides? Ans. 228.8309 and 142.9383.

10. If the sides of a plane triangle be in proportion to each other as the numbers, +, and, what are the angles?

Ans. 117° 16′ 46′′, 36° 20′ 10′′, and 26° 23′ 4′′. 11. In a right-angled plane triangle, the three sides are 3, 4, and 5, it is required to find the angles.

Ans. 36° 52′ 11′′, 53° 7′ 48′′, and 90°. 12. In an oblique-angled plane triangle, the three sides are 4, 5, and 6, what are the three angles? Ans. 41°24′ 34", 55° 46′ 16′′, and 82°49′ 9′′

13. There are three towns, A, B, and c, the distance of A from B is 5 miles, of в from c 9 miles, and of c from a 7 miles, what are their respective bearings from each other?

Ans. A 95° 44′†, Zc 33° 33′‡, ≤в 50° 42′÷. 14. How must three trees, A, B, C, be planted, so that the angle at A may be double that at B, and the angle at в double that at c; and that a line of 100 yards may go just round them?

Ans. The sides are 19.5923, 35.7733, and 44.5945. 15. Suppose a regular pentagon, whose side is 170 fathoms, is to be fortified; and that the salient angle of the bastion is 71°, and its face 47 fathoms; it is required to find the flank and curtain.

Ans. Flank 25.65, Curtain 6457.

OF THE MENSURATION OF HEIGHTS AND

DISTANCES.

The mensuration of heights and distances depends upon the use of certain instruments for taking angles, and the rules of Plane Trigonometry, already delivered; which being separately or jointly applied, as the case may require, will resolve every question of this kind that can occur in practice (p).

In addition, however, to the properties of plane triangles, given in page 10, it may be necessary to lay down a few others relating to angles, parallel lines, &c. which, in several instances, will be found of great use. in facilitating both the constructions and calculations.

1. The two angles, which are made by one right line meeting another, are together equal to two right angles, or 180°.

(p) Horizontal and vertical angles are commonly taken with a theodolite furnished with one or two telescopes, and a vertical arc; and if the circles of the instrument are about 31 inches radius, the observed angles may be read off to half a minute.

But if the angles are oblique to the horizon, they must be taken with a sextant, or Hadley's quadrant, which is held in a position so that its plane may pass through both objects and the eye of the observer and elevations are found by reflecting the object from an artificial horizon.

Short bases, for temporary use only, are usually measured with rods, or the Gunter's chain of 66 feet; but the common 50 or 100 feet tapes are better adapted for expedition. With these lines, when the ground is tolerably level, and the direction, or alignement, of the base pretty correct, the error in distance will proba bly be about 3 inches in 50 feet, or of the whole measurement, as long as the tapes are kept dry.

2. If two right lines intersect each other, the vertical or opposite angles will be equal.

3. A right line intersecting two parallel right lines makes the alternate angles equal; also the outward angle equal to the inward opposite one, on the same side.

4. If one side of a triangle be produced, the outward angle will be equal to the sum of the two inward opposite angles.

5. All angles in the same segment of a circle, or which stand upon the same arc, are equal.

6. An angle at the centre of a circle is double that at the circumference, when they both stand on the same arc. 7. An angle in a semicircle, or that which stands upon half the circumference, is a right angle, or 90°.

8. If a right line be drawn parallel to one of the sides of a triangle, it will cut the other sides proportionally. It may also be remarked, that some of the simplest cases of heights and distances may be resolved without the assistance of trigonometry, or of any instrument for taking angles, by one or other of the following methods:

1. By the property of similar triangles; from which it is known that objects are in proportion to each other as the lengths of their shadows.

C B

Thus, if the height of the pole a c be 8 feet, the length of its shadow cb 6 feet, and the shadow C B, of the object A c, 45 feet:

Then 6 8: 45: 60 feet height a c.

=

2. Another method is by means of two poles of unequal lengths, set up parallel to the object, so that the observer may see the top of the object over the tops of both the poles,

Thus, let the length of the pole de be 5 feet, that of the pole f g 7 feet, their distance asunder eg 8 feet, and the distance ec, of the shorter pole from the ob ject, 180 feet.

Then the triangles dhf and dκ A being similar, dh: hf dк or ec: KA, or 8; 7-5 :: 180: 45 feet AK. Hence AK+KC=AK+de=45+5

50 AC.

3. A third method is by viewing the image of the top of the object reflected from some smooth surface, as a mirror placed horizontally, a vessel of water, &c.

A

B

B

Thus, let в be the reflecting surface, at the distance of 84 feet from the bottom of the object AC; and let a person at D, 7 feet from B, with his eye 5 feet above the ground, view the image of the top of the object at F.

Then, because the triangles BDF, BCA, are similar, it follows, from the principles of optics, that BD: DF: BC CA, or 7 : 54 :: 84: 66 feet = A C.

4. A fourth method, is for the observer to fix a pole upright in the ground, by trials, so that having laid himself on his back, with his feet against the bottom of it, he may see the tops of the pole and the object in the same right line.

D

B

In which case, the distance F D from the foot of the pole to the eye of the observer, will be in proportion to the height of the pole c D, as the whole distance F B is to the height of the object A B.

And if the height of the pole CD be equal in length to the observer F D, the distance FB will be equal to the height of the object A B.

PRACTICAL QUESTIONS.

1. Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, I then found the angle of elevation of its top to be 47° 30'; required the height of the steeple (q).

(9) In Mauduit's Trigonometry (Crakelt's Trans. p. 182.) it is shown that the error of any altitude a c is to the error committed in taking the Abc, as double the height Ac is to the sine of double the observed Abc Whence the error that may arise in taking the said altitude will be the least possible when the sine of double the observed is the greatest possible; which is when it