CHAPTER VI

INACCESSIBLE HEIGHTS AND DISTANCES

THE most correct method of effecting solutions of problems on triangles, such as are given below, is by trigonometry, but the method more generally adopted for ordinary work is to plot the measurements and angles on paper, and to solve the problems by taking measurements from the plan with a scale. The author has considered it unnecessary to include the subject of trigonometry in this elementary work, but as all the questions that can be solved by trigonometry can also be ascertained by plotting on paper, and as in fact trigonometry is more often employed for checking the plottings, its omission will not render the subject of surveying, as treated in this work, incomplete.

EXAMPLE. To find the distance between the two points A and B (Fig. 69), which are inaccessible to each other by reason of the intervening water.

(1) With an Angular Instrument.—Place a dial or theodolite at the point C, whence both stations can be seen. Ascertain the number of degrees in the angle ACB, and measure the two lines AC and CB. The distance can then be found by plotting the angle on paper, marking off the two sides to scale equal to their respective lengths, and measuring the required distance with the scale.

(2) With the Chain alone.-Measure AC and BC and produce the lines to E and D, making CE equal to AC, and CD equal to BC. Then the distance DE being measured will give the required distance between A and B. For the angle DCE is equal to the angle ACB, and the sides CD and CE are

each equal to CB and AB respectively; therefore the two triangles ACB and DCE are equal in every respect, and DE equals AB.

Possibly some obstruction might not allow of the lines AC and BC being produced to D and E, when it would be sufficient to produce the lines on to say a and b. Then when the lines Ca, Cb, and ab were measured, together with the original lines AC and CB, by plotting them, the distance between A and B

could be found. This method is not, however, so accurate as the one previously mentioned, as the farther the lines are produced the more accurate is the work.

(3) With the Cross Staff and Chain.-The cross staff is used by surveyors for laying down lines at right angles to each other. A simple form can be made by taking a threequarter-inch piece of wood, six inches square, and by cutting two grooves along its diagonals, two lines are formed at right angles to each other, which may be used as sights. This is fastened to a staff with a pointed end, so that it can be pushed into the ground with the sights in any required position. To lay down a line at right angles to another, the eye sights along one of the grooves and fixes the staff so that the groove is in

line with the original base line. Then, by looking along the other groove a line can be staffed out at right angles to the first. To return to the example in question.

By means of the cross staff lay down two lines Ac and Bd at right angles to AB, and make AC equal to Bd. Then, if the distance cd be measured, it will be equal to AB, the required distance. As to which of the above methods should be used is a question that would depend upon the circumstances, and to what degree of accuracy the work is required.

EXAMPLE. To find the distance between two objects A and B (Fig. 70), one on each side of a river, without crossing it.

(1) With an Angular Instrument. any line BD, and ascertain its length with the chain. Place the angular instrument at B, and find the number of degrees in the angle ABD; again place the instrument at D, and find the size of the angle BDA. Then by plotting the line BD, the two angles ABD and DBA, and producing the sides of the angles until they meet in a point, the position of A will be fixed on the plan, and its distance from B may be measured with the scale.

From B staff out

A

FIG. 70.

D

E

(2) With the Chain only. -Fix a staff at the point C in line with A and B, another at any point D, and staff out the line AD to E. Measure BC, CE, ED, DB, and CD. Then, by plotting the two triangles CBD and CED in their correct position to scale, the position of A will be fixed on the paper by producing CB and ED until they meet. The distance between A and B can then be measured from the plan. In the above example it would be well to measure BE (on the ground) to serve as a check on the work.

EXAMPLE. To find the distance between two points A and B (Fig. 71), both of which are inaccessible.

(1) With an Angular Instrument.-Staff out any line CD, and measure its length. Place the instrument at C, and ascertain sizes of angles ACD and BCD; again place the instru

ment at D, and ascertain sizes of angles BDC and ADC. Then by plotting the line CD, and also the four angles taken, the position of A and B can be fixed on the plan by producing the sides of the angles, so as to meet in two points, and a measurement with a scale between these points will give the distance between A and B.

(2) With the Chain only.-Staff out any line CD. Place a staff at any point E in line with A and C, a staff F at the point of intersection between two lines ranged out between AD and BC, and still another G at the point in line with EF, and also with BD. Now measure the following lines EC, CD, DG, and GE, the point F being read off in the last line.

Also

measure a diagonal ED, and as a check the other diagonal CG. The quadrilateral figure ECDG can then be plotted, the lines CE and DF produced to meet in a point A, and the lines CF and DG produced to meet in a point B. The distance between A and B can then be measured with a scale.

EXAMPLE. To find the height of a tower AB (Fig. 72), the surface of the land being level.

Place an instrument at a point C some distance from the tower, ascertain the size of the angle formed by the line DA and a horizontal line DF, and measure the distance CB and the height of the instrument. Now plot a line DF of the length measured (DF being equal to CB); draw a line AB at right angles to DF, and from the point D plot an angle ADF equal

Produce the side

to the angle ascertained by the instrument. DA of the angle, and the line BA to meet in A. The height of the tower can then be found by measuring the distance AF on the plan with a scale, and adding on to this length the height of the instrument.

Note. The horizontal line DF is fixed with the spirit level, which is attached to the instrument.

EXAMPLE. To find the height of a tower AB (Fig. 73), the surface of the land not being level.

Place an instrument at any point C, and ascertain the size of the angles formed by a horizontal line, and the lines DA to the top of the tower, and DB to the bottom of the tower; measure the distance DB, and reduce the length to horizontal measure DE (see table, p. 142). Now plot the line DE to

the correct measurement, and draw a line AE at right angles to DE. Also plot the angles ADE and BDE, as ascertained by the instrument, and produce the sides of the angles to meet AE. The height of the tower can then be found by measuring with the scale the length of the line AB.

EXAMPLE. What depth will it be necessary to sink a perpendicular shaft at the point B (Fig. 74) to reach a seam of coal which was found in the shaft AC, the dip of the seam in the direction of AB being given, A and B being in the same level plane? Ascertain the depth of AC, the distance between A and B, and the inclination of the seam in the direction of AB. Plot a horizontal line CE equal to the distance measured between A and B; through E draw a line at right angles to