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In the theodolite, m=29, n=30, and a=30, hence x= 29 x 30 and a-x=30—29'=1', the excess
a space 30 on the limb over a space on the vernier.
Let AB (Pl. 2. Fig. 2), be a portion of the limb of the instrument, and ECD the vernier in one of its positions, its () point coinciding with the line marked 10 on the limb. Now, since the spaces of the vernier are less by l' than the spaces of the limb, the first line on the left of 0, will be l' to the right of the first line on the left of the 10 on the limb, and if the vernier plate be moved l' towards the left, the lines will coincide; when the second line from 0 will be l' to the right of the second line from 10; and if the vernier be moved another minute, these lines will coincide. The vernier would then show 10° 2.
If the vernier plate be turned still farther, till the third, fourth, fifth, &c. lines coincide, it is plain, that the 0 point of the vernier will have passed the line 10 on the limb, by as many minutes as there are lines of the vernier which shall have coincided with lines of the limb. When the last line of the
rnier coincides with a line of the limb, the vernier will have been moved 30, or half a degree; the 0 point will coincide with a line of the limb at the same time, and show 10° 30'.
The general rule for reading the angle for any position of the vernier may now be stated.
When the 0 line of the vernier coincides with a line of the limb, the arc is easily read from the limb; but when it falls between two lines, note the degrees and half degrees up to the line on the right; then pass along the vernier till a line is found coinciding with a line of the limb: the number of this line from the 0 point, indicates the minutes which are to be added to the degrees and half degrees, for the entire angle.
On the vernier of the vertical limb, the 0 point is at the middle division line, and fifteen spaces lie on each side of it, The same relation, however, exists between the spaces of the vernier and those of the limb: the degrees and half degrees are read as before; and for the minutes, we have only to pass along the vernier, in the direction of the graduation of the
limb; and if we reach the extreme line, which is the fifteenth, without finding a coincidence, we must then pass to the other extremity of the vernier, and look along towards the 0 point, till two lines are found which coincide. The lines of this vernier are numbered both ways from the point, and marked 5, 10, 15 to one extremity, and correspondingly from the other extremity 20, 25, 30 to the point again.
The lower telescope being used merely as a guard, requires no adjustment, although it is better to make the axis, about which its vertical motions are performed, horizontal, or perpendicular to the axis of the instrument; and this is easily effected by means of the two small screws k and l, which work into the slide A, that is connected with the horizontal axis.
85. To measure a horizontal angle with the theodolite.
Place the axis of the instrument directly over the point at which the angle is to be measured. This is effected by means of a plumb, suspended from the plate which forms the upper end of the tripod.
Having made the limb truly level, place the 0 of the vernier at 0 or 360° of the limb, and fasten the clamp screw S of the vernier plate. Then, facing in the direction between the lines which subtend the angle to be measured, turn the limb with the outer spindle, until the telescope points to the object on the left, very nearly. Clamp the limb with the clamp screw K, and by means of the tangent screws L and Z, bring the intersection of the spider's lines to coincide exactly with the object.
Having loosened the clamp screw Q, of the lower telescope MN, direct it with the thumb screw P to the same object at which the upper telescope is directed ; then tighten the clamp screw Q. This being done, loosen the clamp screw S of the vernier plate, and with the thumb screws T and Z, direct the telescope to the other object: the arc passed over by the 0 point of the vernier, is the measure of the angle sought.
The lower telescope having been made fast to the limb, will indicate any change of its position, should any have taken place; and, as the accuracy of the measurements depends on examined, and if its position has been altered, the limb must be brought back to its place by the tangent screw L.
It is not necessary to place the point of the vernier at the O point of the limb, previously to commencing the measurement of the angle, but convenient merely; for, whatever be the position of this point on the limb, it is evident that the arc which it passes over is the true measure of the horizontal angle. If, therefore, its place be carefully noted for the first direction, and also for the second, the difference of these two readings will be the true angle, unless the vernier shall have passed the 0 point of the limb, when the greater must be subtracted from 360', and the remainder added to the less.
86. To measure a vertical angle.
The first thing to be done, is to ascertain the point of the vertical limb at which the 0 point of the vernier stands, when the line of collimation of the upper telescope, together with its attached level, is truly horizontal.
If the instrument be accurately constructed, and the parts shall not have been disarranged, this point is the 0 point of the limb. This, however, is easily ascertained by turning the limb till the O's correspond, and then examining if the upper level be truly horizontal. If not, direct the telescope to a distant and elevated object, and read the degrees on the vertical limb. Turn the vernier plate 180', reverse the telescope, direct it a second time to the same point, and read the arc on the vertical limb. The half difference of these two readings, counted from the 0 point of the limb, in the direction of the greater arc read, gives the point of the vertical limb; that is, the point at which the 0 of the vernier stands when the line of collimation is horizontal. This arc is called the correction, and is named minus, when estimated towards the eye-glass, and plus, when estimated in the contrary direction.
These preparatory steps being taken, let the axis of the telescope be directed to any point either above or below the plane of the limb, and read the arc indicated by the 0 of the vernier. If the angle be one of elevation, the correction, when plus, must be added to the arc read on the limb; but if the correction be minus, the difference of the arcs must be taken. When the angle is one of depression, the correction when minus must be added, and when plus the difference between it and the arc read, must be taken. These methods afford the true angles of elevation and depression, and are too obvious to require farther explanation.
87. The preliminary principles being explained, it remains to illustrate the application of trigonometry to the determination of heights and distances.
PROBLEM I. 88. To ascertain the horizontal distance from a point which is inaccessible by reason of an intervening river.
Let C (Pl. I. Fig. 8) be the given point. Measure a horizontal base line AB, such that its extremities, A and B, and the point C, shall not be in the same right line ; and also, so that the point C can be seen from the two points A and B : then measure the horizontal angles, CAB and CBA.
Suppose AB=600 yards, CAB=57° 35' and CBA=64° 51'. The angle C=180°-(A+B)=57° 34'. As sin. C, 57° 34 9.926351 | As sin. C, 57° 34' 9.926351 To AB, 600 2.778151 TO AB, 600 2.778151 So is sin. A, 57° 35' 9.926431 So is sin. B, 64° 51' 9.956744
To BC, 600.11
2.778231 | To AC, 643.49
PROBLEM II. 89. To find the altitude of an inaccessible object, estimated from a horizontal plane passing through a given point.
FIRST METHOD. Let D (Pl. I. Fig. 9) be the inaccessible object, on the top of a hill or tower, and B the point below, through which the horizontal plane is passed; it is required to find the perpendicular DC let fall on the horizontal plane passing through the
Measure any horizontal line for a base, as BA; and the horizontal angles CAB and CBA, as also the angle of eleva
the right angled triangle CBD, there will be known CB and the angle CBD, from which the perpendicular CD may be found (46).
In measuring altitudes and depressions, the height of the limb of the instrument must not be neglected, provided we wish to estimate accurately from the surface of the ground.
Let AB=780, CAB=56" 28, CBA=61° 24' and DBC= 10° 43'. Angle ACB=180°-(56' 28' +61° 24')=62 08'. As sin. ACB 62° 08' 9.946471 | As radius
10.000000 To AB, 780 2.892095 To BC, 735.466 2.866563 So is sin. CAB,
So is tang. DBC
To CB, 735.466 2.866563 To DC, 139.19 2.143606
The angle of elevation may also be measured at the station A, and the altitude calculated, from the horizontal plane passing through that point: a comparison of these altitudes will show the elevation of one extremity of the base line above the other.
SECOND METHOD. 90. Let C (Pl. I. Figs. 10 and 11) be the point in which a perpendicular through the object D, meets the horizontal plane through B. At B measure the angle of elevation DBC. Then having placed a staff at B, and marked on it the exact height of the instrument, measure towards the object any convenient horizontal distance, to a point A, in the direction of the intersection of a vertical plane DBC with the ground. In measuring this line, great care must be taken to keep the chain horizontal; and then it is plain that the sum of the horizontal distances will be equal to the horizontal distance BE, in Fig. 10, or to AE in Fig. 11.
There are three cases :
1. When the point A is above the horizontal plane passing through B.
2. When the point A is below the horizontal plane passing through B; and,
3. When it is a point of the plane.