PRACTICAL GEOMETRY.

By Practical Geometry is commonly understood, the mensuration of heights and distances of terrestrial objects, by means of plane trigonometry.

EXAMPLE I.

In Fig. 1, of Plate 4, is represented a tower AB, of which it is desired to ascertain the heighth above the surface of the ground at B.

Suppose the observer placed at the point C, at the distance CB equal to 130 feet, measured horizontally, or on a level from the foot of the tower at B; his eye being situated at E, elevated 5 feet above the line CB. Then with a common quadrant, or other instrument for measuring angles, let the angle AED be measured equal to 29 degrees, 59 minutes, which is formed by EA the line of sight from the observer's eye to the top of the tower, and the level or horizontal line from his eye to a point D elevated 5 feet above the bottom of the tower at B, corresponding to the elevation of the eye above the ground at C.

This line ED being horizontal, and the face of the tower D being supposed perpendicular to the horizon, the angle ADE will be a right angle; consequently we have a rightangled triangle AED, in which are known the base ED = 130 feet, and the angle at E = 29°, 59, when by Case 3, of right-angled Trigonometry, the perpendicular DE may be found in the following way:

As radius

= 90°, 00' 10,00000

: =

To the tangent of AED 29°, 59′ = 9,76115

But as the point D is elevated 5 feet above the bottom of the tower, this quantity, added to the perpendicular just found 75, will give 80 feet for the whole elevation of the tower, which was required to be known.

Again, by reversing this problem, and supposing that an observer at E found the angle of elevation of the tower AED to be 29°, 59′, and that he knew the height of the tower above the horizontal line ED to be 75 feet, but wished to know the distance between his station and the tower, or the measure of the line ED CB; by the same Case 3,

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of right-angled Trigonometry, he would say;

The angle EAD being the complement of the observed angle AEB to 90 degrees, its measure is 60°, 01', by means of which the horizontal distance of the observer from the tower is found to be 130 feet, agrecably to the statement in the first part of this case, which is applicable to the measurement of all accessible and perpendicular elevations.

EXAMPLE II. Fig. 2, Plate 4.

Suppose it be required to find the height of the point of a pyramid, or obelisk AB, situated on the top of a hill, at the same time that the observer can approach no nearer than the point D, at the foot of the hill. From D let the level, or horizontal line DC, be measured off equal to 360 feet, in such a direction that the obelisk when observed from C may be seen precisely over and in the direction of D. Then, with a proper instrument, let the angle ACD be measured equal to 30°, 00′, as also the angle ADE equal to 46°, 00′.

When

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When these things are ascertained, we have an obliqueangled triangle CAD, of which we know the side CD = 360 feet, the angle ACD 30°, 00', and the angle ADC = 134°, 00' (being the supplement of the measured angle, ADE, to two right angles, or 180 degrees) and, consequently, the remaining angle formed by lines supposed to be drawn from the top of the obelisk to the two stations of the observer, that is, the angle CAD = 16, 00'.

It was shown in Prop. 2 of Trigonometry, that in any plane triangle the sides are in proportion to the sines of the respectively opposite angles; we may therefore by this proportion discover the length of the sides AC or AD. Let. AD then be found as follows:

As the sine of CAD 16, 00' 9,44034

Again, in the triangle ADE, having a right angle at E formed by the imaginary lines DE, which is horizontal, and AE a perpendicular let fall from the point of the obelisk at A, we have the hypothenuse AD just found to be 653 feet, and the angle ADE observed to be 46°, 00'; hence the perpendicular EA may be found as follows, (Trig. Prop. 5.)

But the point E being on the level of the eye of the observer, and supposed 5 feet above the ground line, by adding 5 to 469,7 we obtain 474,7 feet for the elevation of the summit of the obelisk AB, above the plain at the foot of the hill on which it is erected.

EXAMPLE III. Fig. 3, Plate 4.

Suppose it be required to ascertain the length of a flagstaff AB erected on the battlements of a perpendicular tower BC, to the bottom of which we can have access, from the observer's station at the point E. In this case, by measuring the base, or level distance EC, and at E observing the angles formed at the observer's eye by this base, and lines directed to the top and the bottom of the flagstaff at A and B, or AEC and BEC, we can, as in the 1st Example, discover the height BC, and also AC, from which last subtracting BC, the remainder must be the length of the flagstaff AB, which was required. Let, therefore, EC be measured 65 feet, the angle BEC be observed 42°, 30', and AEC be 52, 08 stating this proportion as radius to the tangent of the angle AEC, so is the base EC to the perpendicular height to the top of the flagstaff AC, which will turn out to be 83,6 feet.

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Again, in the triangle BEC, having the angle at E, and the base EC, we discover the height of the tower alone BC to be 59,6 feet; consequently, subtracting this quantity from AC 83,6 feet, the remainder, 24 feet, must be the height of the flagstaff alone, as was required to be discovered.

But on the other hand let it be supposed that the tower is surrounded by a broad ditch, so that the observer can approach no nearer to it than to the point E, then, agreeably to what is said in Example 2, let the horizontal line ED be measured in such a direction that the observer at D

will see the flagstaff immediately over the point E, and let the length of ED be 62 feet.

At E observe the angles formed with the horizon by lines. of sight to the top and the bottom of the flagstaff; the angle AEC being 52°, 08', and BEC being 42°, 30'. In the same way, at D observe the angles ADC = 33°, 21', and

BDC = 25°, 08'.

In the triangle BDE are given the side ED = 62, and the angle BDE 25,08, the angle BED (the supplement of BEC to two right angles) = 137°, 30′, and, consequently, (Geom. Prop. 7,) EBD = 17°, 22': and in the triangle ADE are known the side ED 62, the angle ADE = 33°, 21', the angle AED (the supplement of AEC) = 127°, 52, and, consequently, the angle EAD 18°, 47'. If, therefore, the process pointed out in Example 2 be followed, we shall discover the elevation of the points A and B, the two extremities of the flagstaff, and subtracting the less from the greater of these quantities, the difference will be the length of the staff itself, which was required to be

known.

To find the elevation of the point B:

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