the distance AE between the tree and the station E is 566 yards; the distance DB between the house and the station Dis 640 yards; and the distance BE between the house and the station E is 380 yards.

EXAMPLE VI. Fig. 6, Plate 4.

Let A and B be two objects situated on opposite sides of a rising ground C, so that the one cannot be seen from the other it is required not only to measure the level or hori zontal distance between these objects, but to determine certain points, as E, F, &c. situated directly in the line between them.

Choose any point, as D, from which both objects can be seen, and measure the lines AD and DB, as also the angle formed by these lines, ADB: then in the triangle ADB, the two sides AD and DB being known, and the contained angle, by the 6th Prop. of Trigonometry, the angles BAD and ABD may be discovered, and, consequently, the remaining side AB, which was required.

Again, in order to ascertain points in the line AB, placing a pole, stake, or other moveable object in any direction, as DE, measure the angle formed by this line and DA, that is, the angle ADE: in the triangle DAE, therefore, are known all the angles and one side; the other side DE may of course be found by applying the rules already laid down, which quantity measured off from D to E will determine the position of E in the line of direction between the given objects A and B.

In the same way determine the position of F, and as many other points as may be requisite; when the direction. between the given objects will be ascertained, notwithstanding the high ground which intercepts the view from the one to the other.

EXAMPLE VII. Fig. 7, Plate, 4.

It is required to determine certain points in the line of direction between the objects A and B, which are so situated, that the observer cannot see the one from the other, nor can he conveniently find any one station from which both objects can be perceived.

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The point C is such, that from it the object B may observed; but a hill intercepts the view of the object A: choose, therefore, a position at F, from which may be seen both the object A and the point C. Then measure the lines AF, FC, and CB; also the angles AFC and FCB; and in the triangle FAC, formed by the two lines now measured, and the imaginary horizontal line AC, passing through the intercepting hill, the two sides AF and FC, with the contained angle AFC being known, the remaining angles, and the side AC, will be discovered. Again, in the triangle ACB the side AC being found, and the side CB having been measured, the angle ACB is also known; for it is the difference between FCA, already found, and the measured angle FCB; consequently, all these things being given, the points D, E, &c. in the line AB may be ascertained by the application of the preceding example; because the imaginary line AC has been determined by cal culation, and may be employed as if it had been measured on the ground, and that no obstacle had prevented the view of the object at A from the station of the observer at C.

EXAMPLE VIII. Fig. 8, Plate 4.

Let it be required to measure the perpendicular altitude of a mountain whose summit is at A, by means of two stations B and C at the foot of the mountain.

The line BC elevated to the observer's eye, 5 feet above the ground, is to be measured; then with a proper instruInent measure the angle CBA formed by the line BC from

the

the one station to the other, and the line BA from the station at B to the top of the mountain at A; in the same way the angle BCA is to be measured, when in the triangle BAC knowing the side BC, and the angles ABC and ACB, and of course the angle BAC, the remaining sides BA and CA may be found.

Again, supposing the imaginary line CD to represent the horizon, the angle DCA formed by the horizon and the line CA to the summit of the mountain, may be measured, and AD will represent the perpendicular altitude of the mountain, in which case the angle at D will be a right angle; hence in the right-angled triangle DAC, having the hypothenuse AC and the angle ACD, the perpendicular may be found (Trig. Prop. 5,) to which adding the height of the observer's eye above the surface, the total altitude of the mountain will be ascertained.

EXAMPLE IX. Fig. 9, Plate 4.

Let A, B, and C represent three steeples of a town whose positions and relative distances the one from the other are known, and let an observer at D measure the angles formed at his eye by these objects; it is desired to know how far he is from each of them,

Laying down upon paper, from the plan of the town, the positions of the three steeples, assume any point at pleasure for the observer's station, as at D, through which and the two objects A and B draw a circle: draw lines from D to A and B, and also to C, producing it until it meet the opposite circumference of the circle in E, and draw the lines AE and EB.

From the plan of the town the distances AB, BC, and CA are known, and, consequently, the angles at A, B, and C: the angles ADC ADE and CDB EDB measured by the observer at D are also known. In the triangle BAE

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BAE the side BA is given, and the angle EAB is known, for it is equal to the angle EDB, being angles in the same segment of a circle, (Geom. Prop. 10,) and standing on the same arch or chord EB; hence the two sides AE and EB may be calculated. Again, in the triangle CEB the sides BE and BC are known, as also the angle CBE, which is composed of the angles CBA which was given, and ABE now found, consequently the angle BCE may be calculated, and its supplement to 180 degrees will be the angle BCD, (Geom. Prop. 1.) Then in the triangle DBC the side BC was given, and the two angles BCD and BDC are known, consequently the angle CBD, and the remaining sides BD and CD may be found, which will give the distance of the observer at D from the two steeples at B and C.

Lastly, in the triangle EAC, the sides EA and AC are known, together with the angle EAC, which is composed of EAB already found, and BAC given in the proposition, consequently the angle ECA may be found, the supplement of which to 180 degress will be the angle ACD: then in the triangle ADC are known the angles ACD and CDA, together with the side AC; the side AD may therefore be found, which will give the distance from the observer at D to the third steeple at A.

In the preceding examples mention has often been made of the level or horizontal line; by such a line, in strictness, should be meant one at all parts equally distant from the centre of the earth; but in ordinary language, by a level or horizontal line, is understood a tangent to the earth's sur face at any one place, which, on account of the magnitude of the earth, will, in short distances, have no sensible deviation from the true level or horizon.

Fig. 10 of Plate 4, represents a section of part of the earth, in which C is the centre of the globe, and the circular arch is the surface; CA is a radius, or semidiameter, drawn from the centre to the surface of the sea at the ship A; the dotted line AB is the apparent level or horizontal line of an observer in the ship, which touching the surface at the point A, and produced either towards B or in the opposite direction, falls without the circle, or above the surface; it is, consequently, a tangent at the point A, and at right angles, or perpendicular to the semidiameter, or to the axis of the earth. The point B is the summit of a mountain raised above the true circular surface of the earth at D, and the space DB represents the elevation of the mountain above that surface, or, as it is usually termed, above the level of the sea.

Were the surface of the earth a perfect plain, the tangent at the point A would consequently coincide with every part of that plain, and be likewise a tangent at the point D; but experience shows that this is not the case; for an observer in the ship at A, on approaching the land, discovers first the summit of the mountain at B, and as his distance from the mountain continues to decrease, the more of its elevation does he discover; whereas, had the earth been a plain, the whole elevation BD would have been perceptible at once. It is therefore evident, that as the direction of the tangents to the earth's surface at A and D must be very different, varying in proportion to the distance between the points of contact, the line of sight, or the apparent horizontal level, must also be continually changing its direction; hence the level of the point A would, if produced, extend not to the surface of the sea at the bottom of the mountain at D, but to the summit at B: and hence we have a method of ascertaining the elevation of remarkable mountains, provided we know their distance from the place of observation; as also of determining the magnitude

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