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From this construction, the trigonometrical calculation. is readily deduced. For L-M : M ; ; AB : BE, and L–M : M ; ; AB : BF; whence EF and its half DE, or the radius KE, is found. In like manner, N-H M : M : : CB: BG, and N—M: M. : CB: BH; consequently DI= Barbi In the triangle IBK, the sides BI and BK, with their included angle, are given, and therefore (Prop. 10.) the angle BKI and the base IK are found. Again, all the sides of the triangle IDR being given, the angle IKD (Prop. 14.) is found. Hence, in the triangle ADK, the whole angle AKD and its containing sides are given, and therefore the base AD, or the horizontal distance of the object from the station A is found, and consequently its altitude.—The opposite semicircles will, likewise, by their intersection, give, on the other side, a second position for that point. In practice, the ambiguity would easily be removed.
If the object be seen at the same elevation from all the three points, the arcs of the circles will evidently pass into tangents which bisect at right angles the sides of the triangle ABC. The projection D of the object on the horizontal plane will then be the centre of the circle circumscribing that triangle, and therefore the radius, or the distance AD, will be found by Prop. 18. Book VI. of the Elements.
If the three points of observation should lie in the same straight line, the centres of the determining circles will likewise occur in that line or its extension, and hence the process of calculation will be greatly abridged.
General Scholium. In all the foregoing problems, the angles on the ground are supposed to be taken by means of a theodolite; which, being adjusted by spirit-levels, measures only horizontal and vertical angles, or decomposes other angles into these elements. If the sertant be employed for the same purpose, such angles, when oblique, must be reduced by calculation to their projections on the horizontal plane. In surveying an extensive country, a base is first carefully measured; and the directions of the prominent distant objects being observed from both of its extremities, they are all connected with it by a series of triangles. To avoid, in practice, the multiplication of errors, these triangles should be chosen, as nearly as possible, equilateral.— After a similar method, large estates are the most correctly planned and measured; the ordinary practice of carrying the theodolite with a chain round the boundary being subject to much inaccuracy. If the inequality of the surface of the ground will not admit of the measurement of a base of a sufficient length, a smaller one may be selected at first, and another base derived from this, by combining with it one or more triangles. These triangles, to preclude the multiplication of errors, should be as nearly as possible right-angled, and similar, having their sides increasing in a continued proportion. When this rate of increase is not less than the ratio of the radius to the side of an inscribed equilateral triangle, the number of intermediate triangles between the measured and the computed base will be rather favourable to the accuracy of the result. The vertical angles employed in the mensuration of heights, since they are estimated from the varying direction of the level or the plummet, must evidently, when the stations are distant, require some correction. Let the points A and B represent two remote objects, and C their centre of gravitation ; with the radius CA describe a circle, draw CB cutting the circumference in D and E, and join EA and AD. The converging lines AC and BC will indicate the direction of the plummet at A and B, the intercepted arc AD, will trace the contour of a quiescent fluid, and the tangent AZ, being applied to A, will mark the line of the horizon from that station. Wherefore the vertical angle observed at A is only ZAB, which is less than the true angle DAB, by the exterior angle DAZ. But (III. 21. El.) DAZ being equal to the angle AED in the alternate segment, E. is (III. 15. El) equal to half the angle ACD at the centre. Hence the true vertical angle at any station will be found, by adding to the observed angle half the measure of the intercepted arc ; and this measure depending on the curvature of the earth, which is neither uniform nor quite regular, must be deduced, for each particular place, from the length to the corresponding degree of latitude. Such nicety, however, is very seldom required. It will be sufficiently accurate in practice to assume the mean quantities, and to consider the earth as a globe, whose circumference is 24,856 miles, and diameter 7,912. The arc of a minute on the meridian being, therefore, equal to 6076 feet, the correction to be added to the observed vertical angle must amount to one second, for every 69 yards con
hence easily computed; for (III. 26. El.) AZ*=EZ.ZD, or very nearly ED.ZD; and consequently the visual depression of an object is proportional to the square of its distance AZ from the observer. In the space of one mile, this depression will amount to #: parts of a foot ; and generally, therefore, it may be expressed in feet, by twothirds of the square of the distance in miles. Thus, at twenty miles, the depression is 2663 feet; and at the distance of fifty miles, it amounts to 16663, or nearly the third of a mile. But the effect of the earth's curvature is modified by another cause, arising from optical deception. An object is never seen by us in its true position, but in the direction of the ray of light which conveys the impression. Now the luminous particles, in traversing the atmosphere, are, by the force of superior attraction, refracted or bent continually towards the perpendicular, as they penetrate the lower and denser strata ; and consequently they describe a curved track, of which the last portion, or its tangent, indicates the apparent elevated situation of a remote point. This trajectory, suffering almost a regular inflexure, may be considered as very nearly an arc of a circle, which has for its radius six times the radius of our globe. Hence, to correct the error occasioned by refraction, it will not only be requisite to diminish the effects of the earth's curvature by one-sixth part, or to deduct, from the vertical angles, the twelfth part of the measure of the intervening terrestrial arc. The quantity of horizontal refraction, however, as it depends on the density of the air at the surface, is extremely variable, especially in our unsteady climate.