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CHAPTER VI.

TACHEOMETRY OR STADIA SURVEYING.

Tacheometry.-The term tacheometry is derived from the Greek tacheos (quickly) and metreo (I measure), and therefore signifies the art of rapid measurement. It is, however, now confined to distance and height measurement by telescope.

Principle of the Stadia. its simplest form is supposed to Green in 1778. He used a simple tube containing three horizontal wires, and on the supposition that rays of light travel in straight lines from the object observed to the eye, the distance of the object is proportional to the extent of it apparently intercepted between the cross wires. Thus in Fig. 178, let a be the eye, b, d, c the cross wires, BC and B'c' the extent of the observed objects apparently intercepted by the straight lines Ab, AC produced. If the tube is held so that the line ad from the eye to the central wire is horizontal, then AD and AD' are the distances of the observed objects from the eye, and BC and B'c' being vertical will be bisected by Ad produced in D and D'. The triangles BDA and B'D'A being similar, we have—

AD

AD

The principle of tacheometry in have been first used by Mr W.

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Fig. 178.-Principle of the Stadia.

BD

BD

B

C

BC

B'C'

D'

and as BC and B'C' are respectively double BD and B'D', also

AD

AD'

i.e., the distances from the eye are proportional to the amounts intercepted by the cross wires.

If now the wires are so arranged that when the distance AD is 100 ft., BC is 1 ft., then if B'c' is 2 ft., the distance AD' will be 200 ft., and so on, or in other words the distances will be 100 times the intercepted amounts. Obviously then by placing a staff at c and c', the distances AD and AD' will be equal to the staff readings with the decimal point removed two places to the right.

The horizontal wires at b, c, d are called stadia wires, and the above is the principle of the stadia. The word stadia is Italian, and was originally used to mean the staff. Stadia work is simply another name for tacheometry.

To look at the matter in another way, referring to Fig. 178, we see that the triangles ABC, AB'C' are each similar to the triangle Abc, and we have

If now

AD AD' Ad
BC B'C' bc

Ad
bc

be put equal to K = 100 say, then from (1)—

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Ad

bc

B'C' KX B'C' =

=

= 100 B'C'

(1)

Ad

That is, the ratio being made 100 so that ad is 100 be, then bc the distances AD and AD' are 100 times the intercepted amounts BC and B'c'.

Tacheometry simply consists in applying this principle to the telescope, by means of which the staff may be accurately read at considerable distances.

Distance and Height Measurement by Simple Vertical Angles.-Before describing the tacheometer proper the following method of distance and height measuring by means of simple vertical angles and staff readings may be described :

In Fig. 179, let T be the theodolite and ABC the staff held at c. Let the horizontal distance TD = H be required. Measure the vertical angles DTA and DTB, and let them be 0 and respectively; also let the distance intercepted on the staff be s, being equal to the difference of the staff readings at A and B, which is equal to b-a in Fig. 179.

-:

Now in the figure

b = н tan 0

and a H tan .

Therefore (ba) = s = H (tan
whence-

T

S

tan 0

(3)

66

or the horizontal distance is equal to the difference between the staff readings divided by the difference of the tangents of the observed vertical angles." The computation of H is facilitated by making s an even number = 10 say.

tan

- tan 4)

H

= H

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--

(1)

(2)

Fig. 179.-Distance and Height Measurement by Simple Vertical Angles.

The levels of optical axis of instrument or of staff station are easily found by taking the staff reading at either A or B along with the corresponding vertical component b or a, and treating these as described on page 274.

Bell-Elliott Tangent Reading Tacheometer.-An instrument specially adapted for measuring distances as above described has been patented by Mr G. J. Bell of Carlisle, and is made by Messrs Elliott Brothers, London. The instrument is shown in Fig. 180, and is called the Bell-Elliott tangent reading tacheometer. It is described in Engineering as follows:-"In order to measure distances with this instrument a level staff is erected at the distant point, and the instrument having been carefully levelled, two readings are taken on points of the staff, say 10 ft. apart. Then if be the inclination of the upper line of sight to the horizontal in making the upper reading, and the inclination in

making the lower reading, we have obviously the horizontal distance of the staff

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In Mr Bell's instrument a ready means is provided for reading off these tangents direct from the instrument without referring to tables. Subtracting the two readings from each other, and multiplying the reciprocal of the result by 10 gives at once the horizontal distance. The instrument is essentially an ordinary theodolite fitted with a trough compass, which can be used in the ordinary way. The two attachments which fit it specially for tacheometry consist of, in the first place, a very accurately divided scale rigidly secured to the upper parallel plate, and fitted with a micrometer adjustment, by which it can be shifted with certainty through a distance of 1000 in. up to in. in the direction of its length. This scale can be read through a microscope fixed at right angles to the telescope as shown, and moving with the same, so that its axis makes the same angle with the prime vertical as the latter does with the horizontal plane of the instrument. A total reflection prism is used to deflect the line of vision through the microscope through 90°, so as to bring the eyepiece into a convenient position for use. In making an observation the telescope is first aligned on the upper of the fixed marks on the level staff. The micrometer is then set to zero, and the division of the scale which is nearest to the cross wire of the microscope is read off through the latter. Say this was 19, then the natural tangent of the angle the line of sight makes with the horizontal is 0.19 ± a correction to be obtained from the micrometer head, which is by means of a vernier divided into 500 parts. This head is now rotated till the cross hair of the eyepiece exactly cuts the scale mark, and the reading of the head is then noted. Assume this to be 125. Then the natural tangent of the angle in question is 0.19125. On making similar readings for the lower mark on the staff the numbers might be, say, 17 and 823, then we should have tan 0 - tan $ = 0.01302. The reciprocal of this is 76.805, and the corresponding distance between staff and instrument would be 768.05 ft. This perhaps would be rather a long sight, and the accuracy of the instrument is of course the greater the nearer the staff."

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Fig. 180.-The Bell-Elliott Tangent Reading Tacheometer.

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