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In order to find the distance between two trees A and B, which could not be directly measured because of a pool which occupied much of the intermediate space, I -measured the distance of each of them from a third object c, viz Ac = 588, Bc = 672, and then at the point c took the angle ACB between the two trees = 55° 40'. Required their distance. This is an example to case 2 of plane triangles, in which two sides and the included angle are given. The work, therefore, is left to exercise the student: the answer is 5938

HExAMPLE III.

Wanting to know the distance between two inacees“sible objects, which lay in a direct line from the bottom

of a tower on whose top. I stood, I took the angles of depression of the two objects, viz. of the most remote 25}*, of the nearest 57*. What is the distance between them, the height of the tower being 120 feet? - A The figure being constructed, as in the margin, AB = 120 feet, the altitude of the tower, and AH the horizontal line drawn through its” a-row top; there are given, C old “HAD = 25°30', hence BAp'-BAH – HAD = 64° 30' HAc = 57° 0', hence-BAc = BAH – HAc = 33° 0'. Hence the following calculation.

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"Conseq. BD – BC = 173:656 feet, the distance required. Such is the way in which this problem is usually solved: it may, however, be done more easily and concisely, by means of the natural tangents. For, if AB be regarded as radius, BD and Bc will be the tangents of the respective angles BAD, BAC, and CD the difference of those tangents. It is, therefore, equal to the product of the difference of the natural tangents of those angles into the height AB. Thus, mat. tan'64}^ = -2.0965436 nat, tan 33° = 0-64940.76

difference....... ... 14471360 multiplied by height . . , 120 * *

gives distance cD..... 173-6563200 - t-i- . ...

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ExAMPLE IV.

From the top of a hill I observed two mile-stones on a horizontal road, which ran straight from its bottom, and took their respective angles of depression below the horizontal plane passing through the place of my eye; that of the nearer mile-stone was 14° 3', that of the farther was 3° 56'. Required the height of the hill.

The figure being drawn, it will be found analogous to that in exam. 3, to which, therefore, we shall refer. There are given CD = 1760 yards, the distance between the two mile-stones; ADB = HAD = 3° 56'; AcB = HAc = 14° 3". This admits of three distinct modes of determination, as below.

1st Method.

The angle AcB is equal to the sum of the angles cap and cDA (Euc. i. 32); therefore cAD = AcB – ADB = 10° 7'. Then, in the triangle AcD, it will be, as sin cAD : cil :: sin CDA : cA. And, in triangle AcB, it will be, as rad. or sin B : Ac:: sin c : AB = 166-85; and so, if it be required, is sin cAB : ch = 666-75.

According to this method the logarithmic process will require eight lines. :

2d Method. Since cAD is the difference of AcB and ADC, we have,

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sinc sin D cosec (c - D). . .
Or, making the terms homogeneous (ch. iv.17),
AB. rad} = cd sin c sin D cosec (co- D).
The logarithmic formula is, therefore, this:

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Thus, log co ...... ... 1760. ... 3-2455127
log sin c ...... 14° 3’.. 9-3851924,
log sin D . . . . . . 3° 56' .. 8-8362969.

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This method is, obviously, applicable to all similar 'examples.

3d Method. If AB were radius, cd would evidently be the difference of the tangents of BAD and BAC, or, of the cotancio

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Thus, matcot D = 14.543833 nat cot c = 3'995922

10:547911)1760.000(166.857ft.asbes.
. . . . . . 1054791 -

705209
6328.75

* . 72334
- 63.287
9047
8438
609
527
82
74,

*

—From the comparison of these three methods, it will appear that the secondrought to have the preference to the first:" and that, considering the time employed in looking out the logarithms, the third-method is preferable to the second.

- ExAMPLE V. Wanting to know the height of a church steeple, to the bottom of which I could not measure on account of * a high-wall between me-and-the-chureh, I fixed upon two stations at the distance of 93 feet from each 6ther, on a horizontal line from the bottom of the steeple, and at each of them took the angle of elevation of the top of “the steeple, that is, at the nearest station 55°54'; at the “other 33°20', 'Required the height of the steeple. This is similar to exam, 4, and being worked by the 2a method, the height of the steepio is found to be | 10:27 feet.

ExAMPLE WI.

Wishing to know the height of an obelisk standing at the top of a regularly slophig hill, I first measured from its bottom a distance of 36 feet, and there found the angle formed by the inclined plane and a line from the centre of the instrument to the top of the obelisk 41°; but after measuring on downward in the same sloping direction 54 feet farther, I found the angle formed in like manner to be only 23°45'. . What was the height of the obelisk, and what the angle made by the sloping ground with the horizon?

The figure being constructed, as in C the margin, there are given in the triangle ACB, all the angles and the side AB, to find Bc. It will be obtained by this proportion, as sin c (= 17° 15' = B – A); AB (=54) :: sin A (=23°45') : Bc = 73°3392. Then, in the triangle DBc are known BC as above, BD = 36,

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